Canonical Descriptions of High Intensity Laser-Plasma ...

Canonical Descriptions of High

Intensity Laser-Plasma

Interaction

B. J. Le Cornu

School of Computing, Engineering and Mathematics

University of Western Sydney

A thesis submitted for the degree of

PhilosophiæDoctor (PhD)

2014

http://www.uws.edu.au/scem

http://www.uws.edu.au

Abstract

The problem of laser-plasma interaction has been studied extensively

in the context of inertial confinement fusion (ICF). These studies have

focussed on effects like the nonlinear force, self-focusing, Rayleigh-

Taylor instabilities, stimulated Brillouin scattering and stimulated

Raman scattering observed in ICF schemes. However, there remains

a large discrepancy between theory and experiment in the context

of nuclear fusion schemes. Several authors have attempted to gain

greater understanding of the physics involved by the application of

standard or ‘canonical’ methods used in Lagrangian and Hamiltonian

mechanics to the problem of plasma physics.

This thesis presents a new canonical description of laser-plasma in-

teraction based on the Podolsky Lagrangian. Finite self-energy of

charged particles, incroporation of high-frequency effects and an abil-

ity to quantise are the main advantages of this new model. The nature

of the Podolsky constant is also analysed in the context of plasma

physics, specifically in terms of the plasma dispersion relation. A new

gauge invariant expression of the energy-momentum tensor for any

gauge invariant Lagrangian dependent on second order derivatives is

derived for the first time. Finally, the transient and nontransient

expressions of the nonlinear ponderomotive force in laser-plasma in-

teraction are discussed and shown to be closely approximated by a

canonical derivation of the electromagnetic Lagrangian, a fact that

seems to have been missed in the literature.

Declaration

The work presented in this thesis is, to the best of my knowledge

and belief, original except as acknowledged in the text. I hereby

declare that I have not submitted this material, either in full or in

part, for a degree at this or any other institution.

(Signature)

Acknowledgements

The lion’s share of my gratitude belongs to my principal supervisor,

Dr. Reynaldo Castillo. At any given meeting, Dr. Castillo faced

either a barrage of unfocussed exuberance or grim dejectedness. I

thank him for his years of patience and instruction. My associate

supervisors, Dr. Timothy Stait-Gardner and Prof. Andrew Francis

were instrumental. Prof. Francis first inspired me to pursue science

beyond my undergraduate degree and guided me in my first real foray

into mathematics. Dr. Stait-Gardner was always available to answer

crazy and/or stupid questions, and his lessons were invaluable to me.

All of my supervisors are rare in that they distinguish themselves not

only by the breadth and depth of their knowledge, but by their ability

to impart a genuine understanding of it.

The support of my family as a whole was phenomenal. My father and

my sister Kieren have always been there, supporting me and shielding

me from reality (which often reared its head in bill form.) My mother

and Amro provided the greatest support when it was needed most,

without which I would never have finished this work. Evan and Clare

tolerated my erratic schedule and kept me solvent in the final year,

and Evan perhaps helped keep me sane. Jenna sacrificed a tremendous

amount for me in the first few years, for which I will always be in her

debt. Nidin has been an excellent sounding board and an even better

friend. Thanks also to all my friends who kept me caffeinated and in

contact with the rest of the world, especially Justin and Emma.

Contents

1 Introduction 1

1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Plasma Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2.1 Plasma Frequency . . . . . . . . . . . . . . . . . . . . . . . 10

1.2.2 Debye Length . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.2.3 Collision Frequency . . . . . . . . . . . . . . . . . . . . . . 13

1.2.4 Ohm’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.2.5 Plasma Dispersion Relation . . . . . . . . . . . . . . . . . 16

1.2.6 Refractive Index . . . . . . . . . . . . . . . . . . . . . . . 18

1.3 Fluid Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.3.1 Eulerian and Lagrangian Coordinates . . . . . . . . . . . . 20

1.3.2 Particle-In-Cell Simulations . . . . . . . . . . . . . . . . . 24

1.3.3 Relevant Fluid Equations . . . . . . . . . . . . . . . . . . 24

1.4 Kinetic Plasma Models . . . . . . . . . . . . . . . . . . . . . . . . 25

1.5 Quantum Plasmas . . . . . . . . . . . . . . . . . . . . . . . . . . 27

1.5.1 Scale of Quantum Effects in Plasmas . . . . . . . . . . . . 27

1.5.2 Need For a Quantum Plasma Description . . . . . . . . . . 29

2 Lagrangian and Hamiltonian Mechanics 31

2.1 Lagrangian Mechanics . . . . . . . . . . . . . . . . . . . . . . . . 31

2.2 Canonical Derivation of the Maxwell Energy-Momentum Tensor . 35

2.3 Lagrange Multipliers versus Constrained Variations . . . . . . . . 38

v

CONTENTS

2.4 Canonical Derivation of the Lorentz Force . . . . . . . . . . . . . 42

2.5 Hamiltonian Mechanics . . . . . . . . . . . . . . . . . . . . . . . . 45

3 Lagrangian and Hamiltonian Formulations of Laser-Plasma In-

teraction 52

3.1 The Boltzmann-Vlasov Distribution . . . . . . . . . . . . . . . . . 53

3.2 Ideal Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.3 Clebsch Potential Representation . . . . . . . . . . . . . . . . . . 57

3.4 Virtual Fluid Displacement . . . . . . . . . . . . . . . . . . . . . 63

3.5 Relativistic Constraints in Fluid Dynamics . . . . . . . . . . . . . 69

3.6 Maxwell-Vlasov System . . . . . . . . . . . . . . . . . . . . . . . . 75

3.6.1 Magnetohydrodynamics . . . . . . . . . . . . . . . . . . . 76

3.7 Guiding Centre Motion . . . . . . . . . . . . . . . . . . . . . . . . 78

3.8 The Korteweg-de Vries Equation . . . . . . . . . . . . . . . . . . 80

4 The Energy-Momentum Tensor In Higher-Derivative Theories 82

4.1 The Gauge Invariant Electromagnetic Energy-Momentum Tensor 82

4.2 Lagrangians with Higher-Order Derivatives . . . . . . . . . . . . . 86

4.3 The Gauge Invariant Energy-Momentum Tensor for the Podolsky

Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5 The Nonlinear Ponderomotive Force in Laser-Plasma Interac-

tion 102

5.1 The Ponderomotive Force . . . . . . . . . . . . . . . . . . . . . . 102

5.2 The Nontransient Nonlinear Ponderomotive Force . . . . . . . . . 105

5.3 The Transient Nonlinear Ponderomotive Force . . . . . . . . . . . 110

6 Concluding Remarks 117

References 121

vi

1

Introduction

The study of laser-plasma interaction was born of attempts to recreate something

like the Teller-Ulam Hydrogen bomb on a small, controlled scale. The Hydrogen

bomb is essentially a two-stage device, consisting of a primary fission explosion

that bombards a mixture of Deuterium and Tritium (DT) with enough energy

to create a secondary fusion reaction. The idea to use this same principle for

commercial power generation can be traced to Teller in the late 1950s, although

it was in the early 1960s that Nuckolls, Kidder, Colgate and Zabawski (all of whom

worked at Lawrence Livermore National Laboratory) first considered using lasers

to implode a small DT target, resulting in a fusion reaction [1] [2]. This method

of laser-driven fusion is referred to as inertial confinement fusion (ICF).

Nuclear fusion offers a release of energy per unit mass that cannot be beaten

by chemical or even fission reactions, and fusion energy comes without the cost

of radioactive waste or the threat of nuclear plant meltdown. However, in the

decades since the 1960s, successive predictions of the laser power required for

ignition (the point at which the fusion reaction becomes self-sustaining) have been

defied as the sheer scale and complexity of the physics has gradually been realised.

For instance, one of the many obstacles is the near-perfect spherical compression

of the fusion target that is required to reach ignition. Small irregularities in the

symmetry of the target compression become exponentially more pronounced over

the time scales involved, leading to a loss of energy and momentum delivered to

1

the centre of the imploding target.

The largest inertial confinement facility in the world is the National Ignition

Facility (NIF) located in California. In March 2012, NIF exceeded its design laser

energy output of 1.8 megajoules by delivering 1.85 megajoules in trillionths of a

second - a pulse of more than 500 terawatts at peak power. The ultimate goal of

the NIF is to demonstrate ignition, which has still never been achieved in ICF.

The NIF has failed to meet its own deadline for achieving ignition, but it has

demonstrated a net energy gain from the fusion of a DT pellet compared to the

laser energy that was delivered to it [3] (this is still not a net release compared

to the total laser energy as there are significant losses in the system before the

laser beams reach the target). All theoretical models and simulations indicated

that NIF should be able to achieve ignition comfortably and reasonably quickly

after coming on line [4], and its failure to do so suggests that a sound theoretical

understanding of the problem is still over the horizon.

An alternative to ICF is the magnetic confinement fusion (MCF) scheme.

MCF precipitates a fusion reaction by injecting plasma into a large toroidal

chamber (most commonly a tokamak) - ideally confining the plasma with electro-

magnets arranged around the chamber - while bombarding it with high energy

radiation. However, plasma turbulence is a major problem causing deconfinement

and damage to the reactor interior. Confinement time is usually on the order of

seconds and no MCF facility has achieved ignition to date. The ITER1 (‘the

way’, in Latin) project is in the early stages of construction at the time of writing

and will be the largest MCF tokamak facility in the world upon its predicted

completion in 2019.

The stark disagreement between theory and experiment in ICF and MCF may

simply be due to the enormous number of degrees of freedom that must be taken

into account - too many even for the sophisticated models that have so far been

number-crunched on supercomputers over periods of days or weeks. Alternatively,

1Originally an acronym for International Thermonuclear Experimental Reactor

2

there may be something missing from the theoretical side of physics, lost in

some order of approximation or assumption made for classical physics where

the discrepancy goes unnoticed. Of course, there is no single modification to the

theory of laser-plasma interaction that will serve as a silver bullet. However, it is

the aim of this thesis to present some modifications using ‘canonical’ methods that

may improve the accuracy of theoretical predictions made in the realm of nuclear

fusion. These methods are canonical in the sense that they arise from either a

Lagrangian or Hamiltonian description of the situation. In Lagrangian mechanics,

the equations of motion flow from an application of Hamilton’s principle via a

well-understood process. Hamiltonian formulations may arise from the usual

Legendre transformation of the Lagrangian or a more general transformation

according to Dirac’s constraint theory [5]. Perhaps the most interesting theories

are found by guessing the form of a functional based on the total energy of the

system and its degrees of freedom, entirely independent of a Lagrangian, and then

defining a ‘noncanonical Poisson bracket’ that serves to endow this functional

with a Hamiltonian structure. These canonical methods will be explored in the

context of ICF within this thesis. While nuclear fusion is the main focus of all

research into laser-plasma interaction, the subject is more general than that and

really applies to the physics of any charged particles in the presence of ultra-high

intensity electromagnetic radiation.

There are five principal results presented in this thesis. First, a derivation of

a gauge invariant energy-momentum tensor for a Lagrangian with second-order

derivatives of the electromagnetic potentials is presented [6]. Second, a proof

that the aforementioned tensor reproduces a tensor found by Podolsky [7] for his

theory of generalised Maxwellian electrodynamics is given [8]. Third, Podolsky

electrodynamics is applied to the case of laser-plasma interaction, resulting in

the derivation of a new plasma dispersion relation [8]. Fourth, an expression

closely approximating the nonlinear ponderomotive force, first applied by Hora

[9] to laser-plasma interaction, is derived using Lagrangian mechanics [10]. Fi-

3

nally, a refutation of a result due to Rowlands [11] that attempted to generalise

the transient nonlinear ponderomotive force [12] is put forth, along with a new

alternative generalisation [10].

The remainder of this chapter is devoted to a summary of some basic principles

and equations applied to plasma physics. This summary is not exhaustive, but

is designed to introduce the equations and concepts that will be required later in

a way that ensures this thesis is self contained.

Chapter 2 is a review of all the concepts in Lagrangian and Hamiltonian

mechanics that are necessary for the results presented later. Nothing new is

presented in this chapter, although several derivations are presented that are

not found in standard textbook treatments. These derivations have a twofold

purpose; to familiarise the reader with variational principles and to demonstrate

the general method that will be applied later to achieve new results, so that the

reader can be convinced of their correctness.

Chapter 3 comprises a review of the literature related to the topic of varia-

tional methods in plasma physics and, more generally, in fluid mechanics. The

variational method considered by all these authors is Hamilton’s principle. Some

of the literature specifically addresses the case of laser-plasma interaction, in ei-

ther a Lagrangian or Hamiltonian description, or sometimes both. Within this

chapter, §3.5 contains a critical review of one attempt to apply Hamilton’s prin-

ciple to the case of a relativistic perfect fluid, which does include a minor original

contribution on the part of this author.

Chapter 4 contains the first of the original results presented in this thesis,

namely the derivation of a new gauge invariant energy-momentum tensor for any

gauge invariant Lagrangian dependent on second order derivatives of the elec-

tromagnetic potentials. The Podolsky Lagrangian is then substituted into this

new tensor, yielding an expression that is manifestly gauge invariant without

further tinkering. The equivalence of this expression to Podolsky’s original ten-

sor is then demonstrated. Finally, Podolsky electrodynamics is presented as a

4

1.1 Notation

possible candidate for tackling problems in laser-plasma interaction, due to its

elimination of the infinite electron self-energy and ability to account for higher

frequency phenomena. A new plasma dispersion relation is derived using the

Podolsky equations of motion.

Chapter 5 presents a discussion of the nonlinear ponderomotive force as ap-

plied to laser-plasma interaction. This thesis then points out that derivations that

led to the force expression could probably be replaced by one simple application

of Hamilton’s principle to a slightly modified version of the standard Maxwell

electromagnetic Lagrangian. The transient expression of the nonlinear pondero-

motive force is then briefly discussed, before this author presents an alternate

method of generalising this expression to one that is Lorentz invariant and uses

four-dimensional notation, in contrast to an expression derived by Rowlands [11]

that this author believes to be incorrect.

Chapter 6 contains some concluding remarks that compare the original results

in this thesis with the rest of the literature that was reviewed and presents a

hopeful picture of their application to future experiments in nuclear fusion.

1.1 Notation

All units are SI throughout this thesis. Greek indices will always range from 0

to 3 and Latin indices from 1 to 3 unless otherwise indicated. The summation

convention is employed throughout; a repeated index in any term of an expression

is summed over all possible values, for example,

aiai = a1a

1 + a2a2 + a3a

3.

These indices are known as dummy indices and must always appear in co-

variant/contravariant pairs (respectively lower/upper indices, although there is

no distinction between contravariant and covariant components in 3-dimensional

Cartesian coordinate systems). Free indices appear exactly once in every term

5

1.1 Notation

of an expression, identifying a particular component of a vector or tensor. For

instance, the index i is the free index in the expression ai = gijaj and denotes

the ith component of the vector a. The relationship between covariant and con-

travariant components is determined by the relevant metric tensor gij, so that

ai = gijaj and ai = gijaj. The Minkowski metric used throughout this thesis for

relativistic problems is defined as

gµν =

1 0 0 0

0 −1 0 0

0 0 −1 0

0 0 0 −1

.

The infinitesimal squared spacetime interval is given by

ds2 = dxαdxα = c2dτ 2

where τ is proper time and c the speed of light. The covariant spacetime coordi-

nates are xµ = (ct,−x,−y,−z) which implies that the four-velocity components

are

uµ = dxµ/dτ = (cγ,−vxγ,−vyγ,−vzγ)

where the Lorentz factor is γ = 1/√

1− v2/c2. The covariant components of the

four-gradient are

∂µ =

(∂

∂x0,− ∂

∂x1,− ∂

∂x2,− ∂

∂x3

)=

(∂

c∂t,−∇

).

Comma subscripts always indicate the partial derivative with respect to co-

ordinates xµ, that is,

∂f

∂xµ:= f,µ and

∂f

∂xµ:= f ,µ.

For more complicated tensor expressions, any index appearing to the right of

6

1.1 Notation

the comma denotes a derivative regardless of whether the index is upper or lower

relative to the comma, i.e.

Fαβ,γγ =

∂2Fαβ

∂xγxγ.

The electromagnetic four-potential has covariant components

Aµ = (φ/c,−Ax,−Ay,−Az)

where φ is the scalar potential and A the vector potential. The relationship

between the electric and magnetic fields is given in terms of these potentials such

that

E = −∇φ− ∂A

∂tand B = ∇×A

or, equivalently in component form,

Ei = −φ,i −∂Ai∂t

and Bi = εijkAk,j

where the Levi-Civita tensor is given by

εijk =

1 if ijk = 123, 312 or 231

−1 if ijk = 132, 213 or 321

0 if i = j, j = k or i = k.

The covariant form of the electromagnetic tensor Fµν := Aν,µ − Aµ,ν can be

expressed explicitly in matrix form as

Fµν =

0 Ex/c Ey/c Ez/c

−Ex/c 0 −Bz By

−Ey/c Bz 0 −Bx

−Ez/c −By Bx 0

7

1.2 Plasma Parameters

or, for the contravariant,

F µν = gαµgβνFαβ =

0 −Ex/c −Ey/c −Ez/c

Ex/c 0 −Bz By

Ey/c Bz 0 −Bx

Ez/c −By Bx 0

.

The electromagnetic tensor satisfies the Jacobi identity such that

Fαβ,γ + Fβγ,α + Fγα,β = 0.

The vacuum permittivity and permeability constants are ε0 and µ0 respec-

tively, defined by the relation

c2 =1

µ0ε0.

The Poynting vector, which represents the directed electromagnetic energy

flux density is

S =1

µ0

E×B.

Planck’s constant h in SI units is

h ≈ 6.626× 10−34Js

and the reduced Planck’s constant is

~ =h

2π≈ 1.055× 10−34Js.


While plasma is often referred to as the ‘fourth state of matter’, there is no exact

definition of a plasma. A plasma is best described qualitatively as “an ionized

gas whose behaviour is dominated by collective effects and by possessing a very

8


high electrical conductivity” [13]. There are no exact quantitative criteria that

a gas must meet in terms of temperature, pressure or level of ionisation for it to

be considered a plasma. An ionization level of less than 1% can be sufficient for

a gas to display plasma behaviour. The key distinguishing feature of a plasma

compared to a gas is the collective behaviour of its constituent particles, governed

as they are by the electromagnetic force. In a gas of electrically neutral particles,

interactions are governed by thermokinetics and collisional forces which operate

over shorter distances, essentially at the speed of sound in the gas. The unique

behaviour of a plasma is due to the propagation of electromagnetic influences

throughout an ionised gas at the speed of light.

Plasma physics is a broad discipline that necessarily encompasses fluid dynam-

ics, thermodynamics, relativity and even quantum mechanics, depending on the

mathematical model being employed and the temperatures and densities being

considered. For instance, it has been speculated that Jupiter’s core may consist

of liquid metallic Hydrogen - a theoretical state of matter that can only occur at a

phenomenally high pressure - a state of matter that probably requires a quantum

plasma description. The acceleration of charged particles to near light speeds

in the region of pulsars or active galactic nuclei requires a relativistic plasma

description [14].

If a plasma is modelled as a continuous fluid, then the laws of thermodynamics

are necessary for its complete description. Plasmas are further characterised

according to whether they are ‘hot’, ‘cold’ (high or low degree of ionisation), or

collisional.

Despite the ambiguity in defining a plasma, there are several quantitative pa-

rameters and characteristics that are used to describe only plasmas, aside from the

usual parameters of temperature, pressure, density etc. These will be discussed

in this section.

9


1.2.1 Plasma Frequency

For the purposes of this thesis, the plasma frequency refers specifically to a fre-

quency related to the ionized electrons in a plasma, since their movement within

the plasma is much faster and over much larger distances compared to the sig-

nificantly heavier ions. Exotic plasmas such as electron-positron plasmas or the-

oretical quark-gluon plasmas will not be considered here. Plasmas have a char-

acteristic frequency because the electrons are not really free from the ions even

once stripped from their orbits. The plasma frequency can be derived in several

slightly different ways using: assumptions about the AC shielding of plasmas

[13]; solving the set of equations given by Gauss’ law with the continuity and

Euler equations by linearising them [15, 16, 17], or perhaps more simply from

arguments about perturbations in electron density [18]. The following derivation

follows the same basic argument put forth by Hora [18].

Consider electrons in equilibrium against a static background of positively

charged ions whose mass can be assumed infinite for all practical purposes. If the

electron density is perturbed by a small amount in space, then this variation in

the density is

δn = − ∂n∂xi

δxi (1.1)

and the disturbance creates an electric field where there was none before. Gauss’

law then gives

Ei,i =

q

ε0δn

where q and n are the electron charge and electron number density respectively.

Using Eqn (3.34), this can be expressed as

∂

∂xi

(Ei +

q

ε0nδxi

)= 0.

This implies that the expression in the brackets above is a constant, although

since the perturbation in the density function is creating the electric field (the

10


plasma was assumed neutral to begin with), when δxi = 0 it is also the case that

E = 0 and therefore

Ei = − qε0nδxi. (1.2)

The Lorentz force on an electron whose position was perturbed by the in-

finetismal amount δx when the density was perturbed, is now given by

d2δxi

dt2=

q

mEi.

Using Eqn (1.2), this can be rewritten as

d2δxi

dt2= − q

2n

ε0mδxi

which suggests a solution of the form eiωpt where ωp is identified as the plasma

frequency such that

ωp =

√q2n

ε0m, (1.3)

or, given the best current values for the electron mass, electron charge and electric

permittivity,

ωp = 56.415(m3/2s−1)√n.

The above expression is of course not valid if relativistic effects are enough to

appreciably alter the mass of the electron. The preceding derivation of the plasma

frequency didn’t require any assumptions about or knowledge of a plasma. All

that was referred to was a perturbed density function (although this could be

a density function for any collection of charged particles) and Gauss’ law. This

makes sense, since any electron pushed away from an equilibrium it managed to

reach in the presence of a background electric field will move back and forth in

simple harmonic motion until dissipative forces eventually return it to equilib-

rium.

The plasma frequency is key to understanding how electromagnetic waves

travel through a plasma. When the refractive index of a plasma is derived in

11


§1.2.6, it will become obvious that any light or electromagnetic wave with a fre-

quency less than the plasma frequency will be absorbed or reflected rather than

transmitted by the plasma. Radio waves in certain frequencies can be transmit-

ted between two points on the surface of the Earth without line of sight as the

ionosphere is (as the name suggests) partially ionised and has a characteristic

plasma frequency. The level of ionisation in the upper atmosphere depends on

the time of day, level of solar activity and other considerations.

1.2.2 Debye Length

The Debye length is a fundamental unit of distance over which all significant

plasma interactions are measured. Consider an ion injected into a plasma in

which the electrons and ions are initially equidistant and static. The ion then

creates a disturbance whereby the negatively charged electrons will be attracted

toward the ion and the other ions will be repelled. This results in a cloud of

negative electrons surrounding the positive ion that partially negates the electric

field of the ion, that is, the ion is ‘shielded’ from the rest of the plasma. The

distance over which this shielding occurs is in a plasma is the Debye length. The

derivation of an expression for this length can be done using the equations of

fluid dynamics [13] the balancing of thermal and electric forces [19]. However,

this author puts forward Hora’s [18] much simpler argument for deriving the

Debye length.

Since the plasma frequency provides an inverse time scale over which mean-

ingful phenomena occur within the plasma, the distance over which electrons can

propagate an electrical influence in the plasma is limited by their velocity divided

by the plasma frequency.

According to the law of equipartition [20], the kinetic energy of a particle is

related to the temperature of it species (assuming that the plasma is close enough

to thermal equilibrium to allow a well-defined temperature for its species) such

12


that

1

2m(v2x + v2y + v2z) =

3

2kBT

where kB is Boltzmann’s constant and T is the electron temperature. In terms

of the magnitude of the velocity averaged over three dimensions, this gives

vavg =

√kBT

m.

Therefore the Debye length λD is

λD =vavgωp

=

√ε0kBT

q2n. (1.4)

Collective behaviour unique to plasmas only occurs over distances larger than

the Debye length. Therefore, an important factor to consider in plasma physics

is the number of particles within a certain volume with dimension on the order of

the Debye length. Consider a sphere of radius λD, then the number of particles

Λ within this sphere is given by its volume multiplied by the density function n

[13]:

Λ =4

3πλ3Dn. (1.5)

1.2.3 Collision Frequency

The collision frequency of a plasma refers to the number of collisions per unit

time of particles within the plasma. However, in many circumstances, the electron

temperature (and therefore average speed) is much higher than that of the plasma

ions, and the electron collision frequency is the dominant collision frequency.

A collision is considered to be a strong one when the incident particle’s kinetic

energy is comparable to the Coulomb potential between itself and the particle it

collided with [13]. That is,

1

2mv2 ≈ |qQ|

4πε0rcoul

13


where rcoul is the distance between the particles and Q is the charge on the

particle being collided with, which may be an ion with charge −q or another

electron with charge q. Since only the absolute value is considered, this will be

simplified as |Qq| = q2. If long-range effects are to dominate over short-range

Coulomb interactions, then the average distance between particles in the plasma

ravg must be much greater than the distance at which the Coulomb potential is

comparable to the kinetic energy rcoul. It is common to approximate the velocity

according to the relation

mv2 ≈ kBT,

in contrast to

ravg >>q2

2πε0mv2=

q2

2πε0kBT.

The collision cross section for an electron-electron collision is then given by

σee = πr2coul =q4

4πε20k2BT

2

and the collision frequency must be given by this cross section multiplied by the

particle density and velocity,

νee = σeenv =nq2

4πε20k3/2B m1/2T 3/2

=1

3

ωpΛ.

1.2.4 Ohm’s Law

A version of Ohm’s law can be derived for a plasma using the two-fluid equations.

The following derivation, especially for a collisional plasma, follows Hora [18]. The

two-fluid equations are

minidvidt

= −∇(nikBTi)− ZniqE− niZq

cvi ×B−miniνei(vi − ve) (1.6)

14


for the ion fluid and

menedvedt

= −∇(nekBTe) + neqE + neq

cve ×B−meneνei(vi − ve) (1.7)

for the electron fluid. Recall that since the velocities are for a fluid, the total

time derivative represents the material derivative (see §1.3.1) such that

d

dt=

∂

∂t+ v · ∇v.

The ion and electron fluids will be considered to initially be in a certain state

for which the two fluid equations are exactly solvable. Disturbances created by

waves travelling through the plasma will be ‘perturbed’ solutions to Eqns (1.6)

and (1.7), approximated to first order so that n = n(0) + n(1) and v = v(0) + v(1)

with the first order terms proportional to ei(k·r−ωt). Several assumptions will be

made, namely that the plasma is: collisionless (νei = 0); cold (∇nkBT = 0);

homogeneous (n(1) = 0); without magnetic field (B = 0); and without drift

velocity (v(0) = 0). Eqns (1.6) and (1.7) now become

mini(0)∂vi(1)∂t

= −Zni(0)qE(1) (1.8)

=⇒ imini(0)ωvi(1) = Zni(0)qE(1)

for the ions and

mene(0)∂ve(1)∂t

= ne(0)qE(1) (1.9)

=⇒ imene(0)ωve(1) = −ne(0)qE(1)

for the electrons. Note that Eqns (1.8) and (1.9) combined give

vi(1) = −Zme

mi

ve(1)

and since mi >> me, it follows that the ion velocity is negligible and can be

15


ignored for practical purposes. This leaves the first order equation (1.9) for the

electron fluid velocity, which can be expressed as

ne(0)ve(1) = ine(0)q

meωE(1)

or, recalling Eqn (1.3),

Je = iε0ω2p

ωE(1). (1.10)

Eqn (1.10) is a new generalised Ohm’s law for a collisionless plasma. A col-

lisional version can also be obtained under different assumptions, which will be

discussed later in §5.2. For now, it is enough to give the version of Eqn (1.10)

generalised to include collisions:

dJ

dt+ νJ = ε0ω

2pE(1).

Solving this where J ∝ e−i(k·r−ωt) gives

J =ε0ω

2p

iω(1− iν/ω)E, (1.11)

the collisional version of Ohm’s Law for a plasma.

1.2.5 Plasma Dispersion Relation

Since the dispersion relation for a plasma governs the way in which waves prop-

agate through a plasma, a fluid model of the plasma is necessary to derive it.

Several authors give derivations of the plasma dispersion relation [15, 16, 17],

and while they all use similar arguments, this derivation more closely follows

Somov’s [15].

Consider Eqn (1.10), the generalised Ohm’s Law derived in §1.2.4. Recall

Faraday’s law and Ampere’s Law,

∇× E = −∂B

∂t, (1.12)

16


∇×B = µ0J +1

c2∂E

∂t, (1.13)

and assume the first order electric field contribution E(1) is also proportional to

ei(k·r−ωt), then substitute Eqn (1.10) into (1.13);

∇×B(1) = iµ0ε0ω2p

ωE(1) −

iω

c2E(1).

Taking the time derivative of both sides and then substituting Eqn (1.12)

gives

−∇× (∇× E(1)) =ω2p

c2E(1) −

ω2

c2E(1). (1.14)

Consider any particular component of the left hand side of the above equation,

say, the x-component, then

∇× E = (ikyEz − ikzEy, ikzEx − ikxEz, ikxEy − ikyEx)

=⇒ (∇× (∇× E))x =∂(ikxEy − ikyEx)

∂y− ∂(ikzEx − ikxEz)

∂z

= −kxkyEy + k2yEx + k2zEx − kxkzEz

However, since the wave vector k points in the direction of propagation of the

electromagnetic wave, and the electric field is perpendicular to this direction, it

is the true that

k · E = 0 =⇒ kxEx = −kyEy − kzEz

and therefore

(∇× (∇× E))x = k2Ex.

Substituting this into Eqn (1.14) gives

−k2Ex(1) =

(ω2p

c2− ω2

c2

)Ex(1)

17


and the plasma dispersion relation is therefore

ω2p + c2k2 = ω2. (1.15)

Another way of expressing the dispersion relation serves to highlight the cutoff

frequency below which waves cannot propagate through the plasma;

ck

ω=

√1−

ω2p

ω2.

It is clear from the above expression that any wave with frequency ω ≤ ωp

will have a wave vector that is zero or imaginary. An imaginary wave vector

corresponds to reflection of the wave by the plasma. Since the only variable in

the plasma frequency expression is the density, this problem could be rephrased

to find a critical density of the plasma, above which a wave of frequency ω cannot

penetrate. According to Eqn (1.3),

ne =ε0meω

2p

q2.

Since ω ≥ ωp to avoid being cutoff, the critical density nc can be defined as

nc =ε0meω

2

q2.

If the plasma electron density is equal to or greater than nc, the wave will be

cutoff.

1.2.6 Refractive Index

The refractive index η for any medium is defined by

η =c

vphase

where c is the speed of light and vphase the phase velocity of light in the

18


medium. It is straightforward to derive the refractive index for a collisionless

plasma given the dispersion relation in Eqn (1.15). First, the phase velocity can

be expressed in terms of the frequency and magnitude of the wave vector such

that

vphase =ω

k.

For a plasma, the refractive index is then, from Eqn (1.15),

η =ck

ω=

√1−

ω2p

ω2.

This can be easily adapted for a collisional plasma using the collisional Ohm’s

law in Eqn (1.11). Exactly the same procedure that led to Eqn (1.15) (subsi-

tuting E for J in Ampere’s Law using Ohm’s law and simplifying via Fourier

transformation) can be used to give a collisional dispersion relation using Eqn

(1.11) such thatω2p

(1− iν/ω)+ c2k2 = ω2 (1.16)

and the refractive index is then

η =ck

ω=

√1−

ω2p

ω2(1− iν/ω). (1.17)

It will also be useful later to note that the refractive index is related to the

permittivity and permeability of the medium in question. In free space

c =1

µ0ε0,

and for an electromagnetic wave propagating in a medium with phase velocity

vphase, this becomes

vphase =1√µε

where µ and ε are the permittivity and permeability constants. The auxiliary

fields D and H in matter are then defined in terms of the vacuum electric and

19

1.3 Fluid Mechanics

magnetic fields, E and B:

D = ε0εrE = εE

and

H =1

µ0µrB =

1

µB,

where εr and µr are the dimensionless relative permittivity and permeability

constants, respectively. Therefore,

η =c

vphase=

√µε

µ0ε0=√µrεr. (1.18)

1.3 Fluid Mechanics

Plasma phenomena are typically investigated using one of two broad classes of

models - kinetic models and fluid models. Kinetic models will be discussed briefly

in §1.4. A fluid model describes plasma by looking at its bulk properties as a

fluid, or as two fluids (an electron fluid and an ion fluid) which then allows the use

of all the principles and equations of thermodynamics. A fluid description makes

use of the roughly Maxwell-Boltzmann distribution of particles in collisionless

plasmas. This is obviously a macroscopic idealisation of the physics and such an

approximate description cannot account for certain microscopic phenomena like

plasma double layers.

The material presented in this section is a brief summary of the elementary

concepts and equations of fluid mechanics, which can be found in any textbook

on the subject, e.g., [21] [22].

1.3.1 Eulerian and Lagrangian Coordinates

Fluid mechanics can be described in two equivalent types of coordinate systems -

Eulerian and Lagrangian. In an Eulerian description, the volume under consider-

ation is subdivided into static cells through which the fluid flows. The dynamics

of the system are then described by vectors attached to each cell which change in

20

1.3 Fluid Mechanics

direction and magnitude over time. In a Lagrangian description, fluid particles

are tracked individually as they move throughout a volume over time (one might

practically think of dropping coloured dye or a neutrally buoyant object of negli-

gible mass into the fluid which then stays with a fluid particle as it moves). The

use of the term ‘fluid particle’ and the mathematical description of a fluid as a

continuous system is an approximation. Below the microscopic scale, a fluid is of

course made of molecules and so cannot be defined at every point in space.

An Eulerian position vector will be denoted by x and Eulerian velocity is

v(x, t). The vector position of a fluid particle in the Lagrangian description will

be given by ξ(a, t) where a is the initial position of the fluid particle and t is

time. The initial position a is the label that uniquely identifies each fluid particle

as it moves throughout space. Consistent with this notation, the Eulerian fluid

velocity vE can be given in terms of the Lagrangian particle velocity vL such that

vE(x, t)|x=ξ(a,t) =∂ξ(a, t)

∂t= vL(a, t)

which simply states that the Eulerian velocity at any point in space at a certain

time is the same as the instantaneous velocity of a Lagrangian particle passing

through that point at that time. More generally, any physical quantity carried

along by the fluid must have its time derivative given by

d

dt=

∂

∂t+ v · ∇, (1.19)

which is the material derivative. The material derivative gives the time derivative

holding position constant (the first term) plus the derivative holding time constant

while moving through the vector field (the second term). It is simply the total

derivative of something that depends on both space and time. The acceleration

of the fluid is given by the material derivative of the fluid velocity

dv

dt=∂v

∂t+ v · ∇v.

21

1.3 Fluid Mechanics

If the fluid velocity does not depend on time then it is a steady flow. If it does

not depend on space then the velocity has the same magnitude and points in the

same direction everywhere, which is an isotropic flow (the usage of this term may

differ from other authors). As an example, consider a (fairly unusual) vector field

defined by v = (x + y + t/2, x − y + 2t) where t is a dimensionless parameter

representing time. This vector field is neither steady nor isotropic. This can be

visualised by a stream plot, where a vector is attached to each point in space

indicating the magnitude and direction of the fluid velocity at that point.

-3 -2 -1 0 1 2 3

-3

-2

-1

0

1

2

3

-3 -2 -1 0 1 2 3

-3

-2

-1

0

1

2

3

Figure 1.1: The vector field v = (x + y + t/2, x− y + 2t) at t = 0, 2

The lines formed by the vectors joined tip to tail are referred to as the fluid

streamlines. If the flow were steady, then the streamlines in the plot above on

the left would coincide with the pathlines - the trajectories of Lagrangian fluid

particles. However, with a dependence on time, Fig. 1.1 shows the vector field

drifting away from the origin in the ‘northwest’ direction so the motion of any one

fluid particle is more complicated over time. If a coloured dye were continually

being injected at a certain point, then as the vector field changes in time, the dye

would leave a streakline trailing the leading point of the pathline of the first drop

of dye injected.

To calculate the pathlines of a single particle placed in the fluid, consdier that

22

1.3 Fluid Mechanics

-3 -2 -1 0 1 2 3

-3

-2

-1

0

1

2

3

Figure 1.2: The pathline of a single fluid particle in the steady vector field v =(x + y, x− y)

the velocity vector at any point is parallel to direction in which a particle is in-

finitesimally pushed at that point. So, the expressions for velocity and differential

position vector are

v = (vx, vy),

dr = (dx, dy).

Since these vectors are parallel, it is true that

vxdx

=vydy.

For v = (x + y + t/2, x − y + 2t), the above expression can be integrated to

give

x2

2− y2

2− 2xy + t

(2x− y

2

)= C (1.20)

where C is a constant with values corresponding to the pathlines of different fluid

particles. It can be seen from Eqn (1.20) that in the static case (t = 0), the

pathlines follow a hyperbolic orbit, which is to be expected from looking at the

streamlines in Fig. 1.1 and the pathline in Fig. 1.2. In the unsteady case, as t

23

1.3 Fluid Mechanics

increases, the linear expression 2x− y/2 begins to dominate over the hyperbolic

part of the equation and so the pathlines become flatter and flatter. This can also

be seen by considering that the vector field is drifting away from the origin at a

linear rate, and the further away a particle is from the ‘centre’ of this particular

field, the less curved the streamlines become.

1.3.2 Particle-In-Cell Simulations

‘Particle-in-cell’ (PIC) refers to a method of analysing fluid (and plasma) dynam-

ics. PIC involves solving equations for both the fluid particles in continuous space

using Lagrangian coordinates and the density and current in terms of discrete Eu-

lerian ‘mesh’ points. That is, the continuously variable Lagrangian coordinates

of the fluid particles are overlaid by a grid of cells of finite size, with each cell

corresponding to certain fluid velocity, density etc. This method lends itself to nu-

merical simulations requiring a fairly large degree of computer processing power.

Simulations using a higher density of cells are of course more accurate, but all

simulations are subject to a degree of inaccuracy from the discrete nature of the

Eulerian part of the system.

Numerical simulations using ‘particle-in-cell’ techniques have been explored

since the 1950s [23]. While this thesis focuses on analytical methods of studying

laser-plasma interaction and will not discuss PIC further, it is worth noting that

PIC simulations are still very much an active and fruitful area of research [see

24, 25, 26, 27, 28].

1.3.3 Relevant Fluid Equations

Where there are no gravitational, electromagnetic, or other forces at play, New-

ton’s second law gives the force (mass density n times acceleration) as the negative

gradient of the pressure P ;

ndv

dt= −∇P. (1.21)

24

1.4 Kinetic Plasma Models

Eqn (1.21) is known as Euler’s equation. In general, the fluid density will

depend on space and time, as will the pressure. Recalling Eqn (1.19), Euler’s

equation can also be written as

∂v

∂t= − 1

n∇P − v · ∇v.

Another important equation in fluid dynamics is the continuity equation,

∂n

∂t+ v · ∇(nv) = 0. (1.22)

The continuity equation states that mass is conserved provided there are no

sources or sinks of the fluid in the volume under consideration. The product nv

is referred to as the mass flux density. If there are sources or sinks, then the right

hand side of Eqn (1.22) is nonzero.

If the fluid entropy S is constant in time, then the fluid is said to be isentropic

and

dS

dt=∂S

∂t+ v · ∇S = 0. (1.23)

Analogous to the continuity equation (1.22), the conservation of entropy flux

density can be expressed as

∂nS

∂t+ v · ∇(nvS) = 0.

While fluid mechanics is a vast topic inclusive of complicated subjects like

turbulence, the principles discussed in this section will be sufficient to complement

the discussion of variational principle in plasma physics in this thesis.


Kinetic plasma models tend to rely on numerical techniques involving a large

amount of computational processing, as opposed to fluid models which simplify

25


the situation enough for an analytical study of plasma physics. A kinetic model

may be one in which single particles are individually considered in the plasma,

each governed by the Lorentz force equation. In this case, there is no error due to

assumptions that the plasma is a continuous object in space. The single particle

model is easily the most computationally intensive method of analysing plasma

behaviour, as thousands upon thousands of equations must be solved for each

time step considered in the model if a plasma of realistic size is to be considered.

A kinetic model relying on some distribution function is less accurate than

the single particle model, but still more accurate than a fluid model, provided

the system of equations is solvable. Given a distribution function f(x,v, t), the

probability of finding a particle at time t within an infinitesimal volume x + dx

having velocity in the range v + dv is given by [20]

f(x,v, t)dxdv.

A complete description of the plasma can be found by solving the Boltzmann

or Vlasov equations coupled to the equations of electromagnetism. The Vlasov

equation is [29]

∂f

∂t+ v · ∂f

∂x+ v · ∂f

∂v= 0. (1.24)

Analogous to Eqn (1.19), the Vlasov equation is really just the total derivative

of a function that depends on space, time and velocity, which is equal to zero

when the distribution is conserved in time. If the particles being described by

the distribution function f are charged, then the acceleration can be substituted

by an expression obtained from the Lorentz force such that

∂f

∂t+ v · ∇f +

q

m(E + v ×B) · ∂f

∂v= 0.

The Vlasov equation is really just a collisionless version of the Boltzmann

26

1.5 Quantum Plasmas

equation,

∂f

∂t+ v · ∂f

∂x+ a · ∂f

∂v=

(∂f

∂t

)collisioinal

, (1.25)

where the term on the right hand side of the equation gives the contribution to

the dynamics from collisions of particles (the discovery of the Boltzmann equation

actually preceded the Vlasov equation by more than half a century).

This use of kinetic models will be discussed further in Chapter 3, in the context

of canonical formulations of plasma physics.

1.5 Quantum Plasmas

This section briefly explores the conditions under which quantum effects become

relevant in the description of a plasma. The literature on this subject is fairly ex-

tensive, and the question has been considered by many since the salient principles

of quantum mechanics began to be understood. Among the most notable early

contributions to the understanding of quantum plasmas came from a series of pa-

pers published by Bohm and Pines [30, 31, 32, 33]. Pines also authoured a review

on the subject [34] and several modern reviews are available [see 35, 36, 37].

The question addressed here is: at which densities and temperatures will a

plasma display behaviour that can only be described in the framework of quantum

mechanics? The most important quantum parameters to consider in this case are

the Fermi energy, de Broglie wavelength and spin of plasma particles. In the

case of spin, the magnetization and collision frequency of the plasma determine

whether or not spin contributes in any appreciable way to the plasma dynamics.

1.5.1 Scale of Quantum Effects in Plasmas

Quantum effects are generally thought to be relevant in plasmas of relatively high

density and low temperature. This can be made precise either in terms of the

de Broglie wavelength λD and density n of particles, or in terms of the Fermi

temperature TF = ~2(3π)2/3n2/3/2mkB and thermal temperature T (TFkB is the

27

1.5 Quantum Plasmas

Fermi energy EF ) of the plasma. In terms of these parameters,

nλ3D ≥ 1 OR TF ≥ T

give the transition point at which quantum degeneracy becomes relevant [35].

This is intuitive since if the de Broglie wavelength of particles are larger than

the distance between particles, then it would be expected that they must share

certain quantum numbers (although at least one such number must differ between

them due to the Pauli exclusion principle).

However, the notion that quantum effects are relevant only in relatively high

density/low temperature plasmas was challenged by Brodin et al. [38] who found

that in a two-fluid plasma model in which the electron species is further subdi-

vided into two classes defined by spin, quantum effects are important even in mod-

erate density/high temperature plasmas. Spin flips can be induced in electrons

by collisions or a changing external magnetic field, provided that the magnetic

field varies faster than the inverse electron cyclotron frequency ωce. Therefore,

the time scale t considered in the two-fluid plasma model with electron spin is

1

ωce< t <

1

νe

where νe is the electron collision frequency. In such a case, Maxwell’s equations

in the plasma become

∇ · E =qini − q(n+ − n−)

ε0

∇ ·B = 0

∇×B = µ0 (j +∇× (M+ + M−)) +1

c2∂E

∂t

∇× E = −∂B

∂t

for species of particles a = i,+,− (respectively ions, spin up electrons, spin down

electrons), charge qa, spin sa and magnetization M± = −2µBn±s±/~. The Bohr

28

1.5 Quantum Plasmas

magneton is µB = q~/2me where q and me are the electron charge and mass. The

equations of motion for the plasma particles are

∂na∂t

= −∇ · (nava),

mana(∂

∂t+ va · ∇)va = qana(E + va ×B)− dp

dna∇na

+2µana~

sja∇Bj +~2na2ma

∇

(∇2n

1/2a

n1/2a

),(

∂

∂t+ va · ∇

)sa = −2µa

~B× sa.

Recall that the Einstein summation convention is employed only when dummy

indices appear in upper/lower pairs in each term of an equation; however the

indices s denote only the species of particle and the total expressions require

a sum over all species. The second equation above is the Lorentz force law

with additional terms arising due to the Fermi pressure and spin. Brodin et al.

considered the case of a low frequency Alfven (ion) wave parallel to an external

magnetic field and showed that spin effects are relevant when µB√µ0ρ0/mi ≥ 1,

where ρ0 is the unperturbed density function existing prior to the application

of the external magnetic field. However, spin effects can still be suppressed if

µ0B0/kBTe << 1, in which case the thermal pressure dominates the dynamics.

1.5.2 Need For a Quantum Plasma Description

Experiment at the National Ignition Facility (NIF) in California are currently

charting new territory in condensed matter physics. NIF is capable of studying

matter compressed to the order of thousands of times that of lead under normal

conditions and at pressures on the order of Tera Pascals. The properties of matter

at such high densities are not well understood but is important to astrophysicists

as much as nuclear fusion scientists. Metallic Hydrogen is created at pressures

high enough to force the atoms to occupy space within each other’s Bohr radii,

and is thought to exist in the cores of Jupiter and Saturn. In 1996 physicists from

29

1.5 Quantum Plasmas

Lawrence Livermore National Laboratory (NIF is also located inside the LLNL)

reported the creation of metallic Hydrogen [39] for roughly a microsecond. Most

recently, Eremets and Troyan reported the creation of liquid metallic Hydrogen

and Deuterium below 300 GPa [40]. The quantum properties of the most basic

element at high densities must be understood by physicists in general, and may

be important in developing inertial confinement fusion.

A 2011 report emerging from a National Nuclear Science Administration and

Office of Science workshop highlights, among other things, the theoretical predic-

tions regarding dense matter that are yet to be confirmed but which theoretically

lie within reach of the NIF [41]. These predictions include: the creation of solid

metallic Hydrogen; a plasma phase transition in the fluid phase for Hydrogen and

Helium; a maximum of the melting curve; the melting of Hydrogen at T = 0 K;

a Wigner crystal state for Hydrogen; a superconductor and/or superfluid phase

of Hydrogen. A better theoretical understanding of the nature of quantum plas-

mas will be crucial in any further study of the unique properties of ultra-high

condensed matter.

30

2

Lagrangian and Hamiltonian

Mechanics

2.1 Lagrangian Mechanics

This chapter introduces the basic principles of Lagrangian and Hamiltonian me-

chanics. A review of the application of these ideas to the case of fluid mechan-

ics and laser-plasma interaction will be presented in Chapter 3. Discussions of

Hamilton’s principle and the Euler-Lagrange equations can be found in any good

classical mechanics textbook [e.g. 42, 43].

Lagrangian mechanics is a desirable framework in which to study plasma dy-

namics (and indeed many other areas of physics) due to the natural appearance of

the equations of motion and conservation laws from well understood operations on

a single functional - the Lagrangian. The application of Lagrangian mechanics to

plasma physics originated with Low’s Lagrangian formulation of the Boltzmann-

Vlasov equation [44]. The most notable extension of this theme in describing the

complexities of plasma physics was the noncanonical Hamiltonian formalism used

by Littlejohn in the context of guiding centre motion in magnetic confinement

fusion [45, 46]. Much work has been done in this field since these early efforts

and Hamiltonian and Lagrangian mechanics are of continuing interest to those

31


studying plasma physics and fluid dynamics in general [see 28, 47, 48, 49].

Lagrangian mechanics is an elegant formulation of classical mechanics that

allows the equations of motion to be derived from the principle that the path an

object takes will always be the path that gives a stationary value of the action.

This is known as Hamilton’s Principle, or sometimes the Principle of Least Action

(although this is a misnomer, as the action need only be an extremal value, not

necessarily a minimum). The action is a functional with units of J · s given by

S =

∫Ldt

where L is the Lagrangian, which in classical mechanics usually a function of

some generalised coordinates qi and their first derivatives with respect to time, qi.

When a Lagrangian system on a Riemannian manifold is the difference between

kinetic and potential energy of a system it is called natural and corresponds to a

mechanical system [50]. However, the Lagrangian is not necessarily a physically

meaningful quantity; it is not restricted by demands on measurability or gauge

invariance, for instance.

Hamilton’s Principle states that the path followed by the system will be the

one which extremises the action. The equations of motion then come from the

requirement that the Action be an extremum value for the path taken between

two points t1 and t2 in time. In functional calculus, this is equivalent to saying

that the variation of the action must be 0;

δS =

∫ t2

t1

δLdt =

∫ t2

t1

(∂L

∂qδq +

∂L

∂qδq

)dt = 0. (2.1)

A variation with respect to time is not explicitly included since the two points

t1 and t2 are fixed. The variations in the path are required to be 0 at the end

points, so that any path taken must at least start and end at t1 and t2 respectively,

which means that δq(t1) = δq(t2) = 0.

32


Integrating the second term in Eqn 2.1 by parts and using δq = dδq/dt gives

∫ t2

t1

(∂L

∂q− d

dt

(∂L

∂q

))δqdt = 0

and since this must hold for arbitrary variations δq, it must be true that

d

dt

(∂L

∂q

)=∂L

∂q. (2.2)

This is the Euler-Lagrange equation, but of course for n generalised coordi-

nates we have n equations. The Euler-Lagrange equations are in covariant form,

that is, they have the same functional form under any invertible transformation to

a new set of generalised coordinates q(q, q) and ˙q(q, q). More generally, the Euler-

Lagrange equations come from finding the stationary points of S with respect to

a functional derivative

δS

δq= 0,

completely analogous to the way in which stationary points of functions are found.

A specific example will serve to better illustrate the usefulness of the Euler-

Lagrange equations. Consider the Lagrangian for a simple one-dimensional par-

ticle with mass m in a field,

L(x, x) =1

2mx2 − V (x),

where V is the potential energy of the field which is assumed to not change with

time. The Euler-Lagrange equation 2.2 gives

d

dtmx = −∂V

∂x,

which is instantly recognisable as Newton’s second law, F = ma. In fact, the

most well-known physical equations can be found via this procedure given an

33


appropriate Lagrangian. The Lagrangian density

L = − 1

4µ0

FαβFαβ − JαAα

yields Maxwell’s inhomogeneous equations in the form

F µα,α = −µ0J

µ.

The relativistic Lorentz force,

d

dtmvγ = q(E + v ×B),

is found via the Lagrangian

L = −mc2γ−1 − qφ+ qv ·A

(see §2.4 for this derivation). The Klein-Gordon equation (relativistic Schrodinger

equation), (∂α∂

α +m2c2

~2

)ψ = 0,

comes from the Lagrangian

L = −mc2ψψ − ~2

mψ,αψ

,α.

The Dirac Equation, which describes the wavefunction of spin-1/2 particles,

(iγα∂α −

mc

~

)ψ = 0,

can be found from the Lagrangian

L = −mc2ψψ + i~cψγα∂αψ.

34

2.2 Canonical Derivation of the Maxwell Energy-Momentum Tensor

where γα in the Dirac equation and its Lagrangian represents the four gamma-

matrices [51], not the relativistic Lorentz factor used in the Lorentz force expres-

sion. The complex conjugate of ψ is represented by ψ.

The power of Hamilton’s principle should be clear; the equations of motion

for physical systems can be derived without a priori knowledge of them, provided

that the correct form of the Lagrangian is known. It was already mentioned that

the Lagrangian is often the difference between the kinetic and potential energy of

the system in classical mechanics, and this fact can be used as a guiding principle

in constructing more complicated Lagrangians for new systems.

2.2 Canonical Derivation of the Maxwell Energy-

Momentum Tensor

Recall Eqn (2.2) which was derived from the Principle of Least Action. Using

this standard form of the Euler-Lagrange equations for a Lagrangian L dependent

on some generalised coordinates q and their first derivatives with respect to time

only. Of course, in relativistic theories, the time coordinate is on the same footing

as the space coordinates and the Euler-Lagrange equation is usually expressed as

d

dxµ

(∂L

qi,µ

)=∂L

∂qi.

The total derivative of the Lagrangian with respect to the spacetime coordi-

nates can then be expressed as follows:

dL

dxµ=∂L

∂qiqi,µ +

∂L

∂qi,νqi,νµ +

∂L

∂xµ

=d

dxν

(∂L

∂qi,ν

)qi,µ +

∂L

∂qi,νqi,νµ +

∂L

∂xµ

=d

dxν

(∂L

∂qi,νqi,µ

)+∂L

∂xµ

=⇒ − ∂L

∂xµ=

d

dxν

(∂L

∂qi,νqi,µ − δνµL

)

35


Note that in the preceding derivation, the indices following a comma represent

a total derivative of the generalised coordinates. This was done for notational

simplicity but it is generally clear from the context of a particular problem as

to whether a total or partial derivative is called for. In this way, the partial

derivative of the Lagrangian with respect to the spacetime coordinates (usually

the gradient of the potential fields which is the force) is expressed in terms of

the four-divergence of a tensor which shall be referred to here as the canonical

energy-momentum tensor T νµ

T νµ =

∂L

∂qi,νqi,µ − δνµL. (2.3)

The Maxwell stress tensor is derived from the Lagrangian

L =ε02E2 − 1

2µ0

B2

using the canonical definition of the energy-momentum tensor T nm with respect

to the generalised coordinates φ and Ai such that

T nm =

∂L

∂φ,nφ,m +

∂L

∂Ai,nAi,m − δnmL. (2.4)

This gives

T nm = ε0φ

,nφ,m + ε0∂An

∂tφ,m −

1

µ0

εinqεilmAm,lAq,m + cε0δ

n0φ

,qAq,m + cε0δn0

∂Aq

∂tAq,m

− δnmL

= −ε0Enφ,m −1

µ0

εinqBiAq,m − cε0δn0EqAq,m − δnmL

Taking the divergence and rearranging some terms,

36


T nm,n =

∂

∂xn(ε0E

nEm − δnmL) + ε0EnAm,n∂t− 1

µ0

εinqBi,nAq,m −1

µ0

εinqBiAq,mn

− ∂ε0EnAn,m∂t

=∂

∂xn(ε0E

nEm − δnmL) + ε0EnAm,n∂t

+ ε0∂En

∂tAn,m −

1

µ0

BnBn,m −

∂ε0EnAn,m∂t

=∂

∂xn(ε0E

nEm − δnmL) + ε0En∂Am,n − An,m

∂t+

1

µ0

Bn(B ,nm −Bn

,m)

− 1

µ0

BnB,nm

=∂

∂xn(ε0E

nEm −1

µ0

BnBm − δnmL) + ε0En∂Am,n − An,m

∂t− 1

µ0

Bn(B ,nm −Bn

,m)

+2

µ0

Bn(B ,nm −Bn

,m)

=∂

∂xn

(ε0E

nEm +1

µ0

BnBm − δnm(ε02E2 +

1

2µ0

B2

))+ ε0E

n∂Am,n − An,m∂t

− 1

µ0

Bn(B ,nm −Bn

,m)

In vector notation, this becomes the familiar expression for the force in terms

of the divergence of the Maxwell stress tensor U, plus the time derivative of the

Poynting vector S,

f = ∇ ·U− 1

c2∂S

∂t,

such that

f = ∇ ·(ε0E⊗ E +

1

µ0

B⊗B− 1

2I

(ε0E

2 +1

µ0

B2

))− ε0

∂E×B

∂t(2.5)

where I is the 3×3 identity matrix. Therefore, it is not just the equations of mo-

tion that can be found in a Lagrangian framework, but also the conservation laws

of the system. This idea is expressed more generally in Noether’s theorem, which

states that for every ignorable generalised coordinate in the Lagrangian, there is

a corresponding conserved quantity [52]. For instance, a Lagrangian system that

does not depend on time is one that conserves energy. A Lagrangian system that

is invariant under any translation of the coordinates conserves momentum. Ro-

37

2.3 Lagrange Multipliers versus Constrained Variations

tational invariance implies conservation of angular momentum, and so on. The

power of Noether’s theorem is that it can be applied to any set of generalised

coordinates to find the corresponding conserved quantities in the system, which

are often not obvious from the nature of the problem.

2.3 Lagrange Multipliers versus Constrained Vari-

ations

The above description of Hamilton’s principle is a special case where the coor-

dinates q and q are varied independently of each other (and no second order or

higher derivatives of q appear in the Lagrangian). In many problems of physical

interest, there are constraints that must be taken into account. A constraint that

depends only on the generalised coordinates (and possibly time) is referred to as

a holonomic constraint. A constraint that depends on the velocities is nonholo-

nomic or anholonomic. Such constraints are path-dependent and non-integrable.

There are two equivalent ways in which constraints can be introduced into

Hamilton’s principle. The first method of introducing a constraint on the gener-

alised coordinates and/or velocities is via a Lagrange multiplier. Lagrange multi-

pliers are familiar to anyone with a basic knowledge of maximising and minimis-

ing problems in calculus. As an example, consider the free space electromagnetic

Lagrangian density,

L = − 1

2µ0

B2 +ε02E2.

Attempting to retrieve Maxwell’s equations from Hamilton’s principle would

yield trivial solutions if E and B were varied independently. Of course, in elec-

tromagnetism the electric and magnetic fields are constrained by their expression

38


in terms of the electric and magnetic potentials φ and A such that

E = −∇φ− ∂A

∂t,

B = ∇×A.

If the equations of motion,

∇ · E = 0,

∇×B− 1

c2∂E

∂t= 0,

are known a priori, then they can be introduced to the Lagrangian with the aid

of a Lagrange multiplier. Hamilton’s principle would then yield the expressions

for E and B in terms of the potentials. This can be seen with the following

Lagrangian dependent on the electric and magnetic fields, as well as four new

variables in the form of the Lagrange multipliers λi and α such that

L = − 1

2µ0

B2 +ε02E2 + λ ·

(1

c2∂E

∂t−∇×B

)+ α∇ · E.

The six Euler Lagrange equations (technically ten - three components of E

and B each as well as four for the Lagrange multipliers which simply return the

constraints) then give

∂L

∂E− ∂

∂t

∂L

∂ ∂E∂t

− ∂

∂xi∂L

∂ ∂E∂xi

= 0 =⇒ E =1

ε0∇α + µ0

∂λ

∂t,

∂L

∂B− ∂

∂t

∂L

∂ ∂B∂t

− ∂

∂xi∂L

∂ ∂B∂xi

= 0 =⇒ B = −µ0∇× λ.

Therefore, Hamilton’s principle returns the constraints on E and B in terms

of the potentials where the Lagrange multipliers are revealed as

λ = − 1

µ0

A and α = −ε0φ.

39


Alternatively, if the constraints on E and B in terms of the potentials were

known a priori, then the equations of motion could be derived from the La-

grangian

L = − 1

2µ0

B2 +ε02E2 + λ · (B−∇×A) + α ·

(E +∇φ+

∂A

∂t

)

where there are now six new variables corresponding to the three components of

the vectors λ and α.

The second method of dealing with constraints is the method of constrained

variations where the Lagrangian is unchanged, but the variations of the coordi-

nates in Hamilton’s principle are done with respect to the constraints. Continuing

the use of electromagnetism as an example, assume that the coupling between E

and B is known so that the variations δE and δB are connected via the potentials

φ and A such that

δE = −∇δφ− ∂δA

∂t,

δB = ∇× δA.

The Lagrangian itself remains unchanged as

L = − 1

2µ0

B2 +ε02E2,

but the application of Hamilton’s principle now looks like

δS =

∫ (∂L

∂E· δE +

∂L

∂B· δB

)d3xdt

=

∫ (∂L

∂E·(−∇δφ− ∂δA

∂t

)+∂L

∂B· ∇ × δA

)d3xdt

=

∫− ∂

∂t

(∂L

∂E· δA

)−∇ ·

(∂L

∂B× δA +

∂L

∂Eδφ

)− δφ∇ · ∂L

∂E

+ δA ·(∂

∂t

∂L

∂E+∇× ∂L

∂B

)d3xdt = 0.

40


The independent variations of φ and A then give Maxwell’s equations,

δφ : ∇ · E = 0,

δA :1

c2∂E

∂t−∇×B = 0.

Interestingly, this constrained variational principle has also yielded some extra

terms in the integrand in the form of a four-divergence (partial derivative of time

plus divergence in space). This suggest that Noether’s theorem can be applied to

also retrieve information about the conserved quantities in the system. Energy

conservation is given by considering infinitesimal time displacements δt such that,

in this case,

δA =∂A

∂tδt and δφ =

∂φ

∂tδt.

The remaining terms in the varied action integrand (now varied with respect

to δt),

− ∂

∂t

(∂L

∂E· ∂A

∂t

)−∇ ·

(∂L

∂B× ∂A

∂t+∂L

∂E

∂φ

∂t

)= 0,

give

1

2

∂

∂t

(−ε0E2 +

1

µ0

B2

)= 0.

In free space, this is the energy conservation law

1

2

∂

∂t

(−ε0E2 +

1

µ0

B2

)= −1

2

∂

∂t

(ε0E

2 +1

µ0

B2

)−∇ · S = 0.

Considering infinitesimal spatial displacements δx such that

δA = (δx · ∇)A and δφ = (δx · ∇)φ

gives the conservation law with respect to translational invariance, which is the

momentum conservation law for the electromagnetic field in free space,

∇ ·U =1

c2∂S

∂t,

41

2.4 Canonical Derivation of the Lorentz Force

where U is the Maxwell stress tensor derived previously in §2.2. The same proce-

dure shown above can be repeated for constrained variations in terms of Maxwell’s

equations, which would yield expressions for the electric and magnetic field in

terms of the potentials.

In this section, the problem of constraints on a Lagrangian system has been

dealt with using the well-known example of the electromagnetic field equations.

In both the Lagrange multiplier and constrained variation methods, it was shown

that knowledge of one set of constraints - either the form of the equations of

motion or the relationship of the vector fields with the potentials - yielded in-

formation about the other set of constraints. While the results are the same for

either method, it is this author’s opinion that the constrained variation approach

is slightly superior in that the conservation laws are also made immediately made

apparent by Hamilton’s principle. In the case of Lagrange multipliers, slightly

more working would be required to derive the energy-momentum tensor and ap-

ply Noether’s theorem. However, it is also worth noting that the constrained

variation approach was slightly more algebraically complex in this particular ex-

ample.

Both the constrained variation method and Lagrange multipliers are used by

authors whose work is reviewed in Chapter 3.


The Lorentz force,

f = q(E + v ×B),

can also be derived using a Lagrangian formalism (note that some authors refer

to just the v×B term as the ‘Lorentz force’ but this thesis uses the name for the

total force). The total Lagrangian for a charged particle in an electromagnetic

field is

L(Aµ, Aµ,ν , ξi, ξi) = − 1

4µ0

FαβFαβ − JαAα +mc2(γ − 1) (2.6)

42


where Jα = ρ0ξα is the four current and ρ0 is the rest frame charge density. Note

the use of the label ξi for the particle coordinates, instead of the usual xi. This

is done to highlight that the dynamic variables ξi refer to the particle position, a

very different concept to the fixed field coordinates xµ on which the generalised

coordinates Aµ depend. This is a subtle point that is often not addressed by

other authors, but the situation is analogous to the difference between Eulerian

and Lagrangian fluid coordinates discussed in §1.3.1. Both the four velocity and

Lorentz factor γ can be expressed in terms of just the three coordinates ξi.

The term on the left in (2.6) is the pure field term, while the term on the right

is the pure particle term (kinetic energy of the particle). The middle term gives

the interaction between the particle and the field. Note that the kinetic energy

of the particle is given by the total relativistic energy E minus the rest energy

mc2 where

E =√m2c4 +m2v2γ2c2 = mc2γ

and v = ξ is the particle velocity. The reason for the addition of the constant

rest energy to the Lagrangian will become clear in a moment.

The Euler-Lagrange equations with respect to the generalised coordinates Aµ

in (2.6) give the electromagnetic field equations of motion (discussed at length

in §2.3), while the Euler-Lagrange equations with respect to ξi give the particle

equations of motion. To derive the Lorentz force, only the particle and interaction

terms depend on ξi, so only these two terms are required such that

L(ξi, ξi) = −JαAα +mc2(γ − 1). (2.7)

For the Lagrangian in (2.7), the action would be given with respect to proper

time τ such that

S =

∫Ld3xdτ,

but a Lagrangian expressed in vector notation can be found given that γdτ = dt

43


so that

S =

∫(−γ−1JαAα −mc2(γ−1 − 1))γd3xdτ =

∫(−qφ+ qv ·A−mc2γ−1)d3xdt,

where the constant rest energy has now been dropped from the left hand side,

and a Lagrangian equivalent to (2.7) is therefore

L = −qφ+ qv ·A−mc2γ−1. (2.8)

Plugging the Lagrangian from (2.8) into the Euler-Lagrange equations gives

∂L

∂ξi=

d

dt

(∂L

∂vi

)−qφ,i + qvjAj,i = q

dAidt

+mdviγ

dt

−qφ,i − q∂Ai∂t

+ qvjAj,i − qvjAi,j = mdviγ

dt

Note that ξ has now been identified with xi in the derivatives of the potentials

since the particle position within the field must be considered in the interaction

term, analogous to the way in which an Eulerian velocity vector field at a certain

point in space is identified with the Lagrangian fluid particle velocity at that

point in time. Using the definition of the electric and magnetic fields where

E = −∇φ− ∂A

∂tand B = ∇×A,

the Lorentz force can be expressed in vector notation now as

dmvγ

dt= q(E + v ×B).

Since the Lorentz force is derived only from the Lagrangian terms correspond-

ing to the kinetic energy of the particle and the interaction term, it is clear that

44

2.5 Hamiltonian Mechanics

adding non-standard field terms to the Lagrangian will have no effect on the

Lorentz force. This will be crucial in §4.3 where the Podolsky Lagrangian - a

non-standard Lagrangian for electrodynamics - is applied to the case of laser-

plasma interaction.


Hamiltonian mechanics was developed as an alternative to the Lagrangian formu-

lation of classical mechanics. While in classical mechanics the Lagrangian is the

difference in kinetic and potential energy of a system, the Hamiltonian is their

sum, equal to the total energy of the system. The Hamiltonian description is

usually not easier to work with or develop than the Lagrangian description, but

it has many edifying qualities not found in Lagrangian mechanics. A Hamiltonian

gives the equations of motion for 2n variables in terms of first order differential

equations, as opposed to the Euler-Lagrange equations which are second order

differential equations for n generalised coordinates. The true value of Hamilto-

nian mechanics in modern physics lies in its applicability to quantum mechanics

where ‘canonical quantization’ of a system converts the total energy into a Hamil-

tonian operator that generates time evolution of the system. For instance, the

time-dependent Schrodinger equation is expressed as

i~∂ψ

∂t= Hψ

where ψ is the wavefunction of the system and H the Hamiltonian operator,

found by canonical quantization of its coordinates and momenta. Without a

well-defined Hamiltonian, a quantum description of a system is not possible.

Textbook descriptions of Hamiltonian mechanics usually begin with a La-

grangian function before proceeding to the Hamiltonian H(p, q, t) via a Legendre

transformation,

H(p, q, t) = piqi − L(q, q, t) (2.9)

45


where the canonical momentum p is

pi =∂L

∂qi. (2.10)

The derivation of Hamilton’s equations is elementary and found in any good

textbook on classical mechanics, e.g. [42]. Consider the differential of the Hamil-

tonian,

dH(q, p, t) =∂H

∂qdq +

∂H

∂pdp+

∂H

∂tdt

=∂(pq − L)

∂qdq +

∂(pq − L)

∂pdp+

∂(pq − L)

∂tdt

= − d

dt

(∂L

∂q

)dq + qdp− ∂L

∂tdt (using 2.2)

= −pdq + qdp− ∂L

∂tdt.

Equating coefficients of the differentials dq, dp and dt in the first and last lines

above gives Hamilton’s equations:

∂H

∂q= −p, ∂H

∂p= q,

∂H

∂t= −∂L

∂t. (2.11)

Given any arbitrary density function in phase space f(q, p, t), its total time

derivative is

df

dt=∂f

∂t+∂f

∂qiqi +

∂f

∂pipi.

If Hamilton’s equations are used to replace q and p, then

df

dt=∂f

∂t+∂f

∂qi

∂H

∂pi− ∂f

∂pi

∂H

∂qi=∂f

∂t+ {f,H}

where the Poisson bracket is

{f, g} :=∂f

∂qi

∂g

∂pi− ∂g

∂qi

∂f

∂pi. (2.12)

46


If the density function f obeys Liouville’s theorem then df/dt = 0 [see 20,

Ch. 1, §3] and the time evolution of f is given by

∂f

∂t= −{f,H}. (2.13)

Indeed, it is clear that Hamilton’s equations for q and p are also expressible

in terms of the Poisson bracket

q = {q,H}, p = {p,H}, (2.14)

and this is the essential feature of a Hamiltonian theory - the time evolution of

any quantity in phase space is said to be generated by the Hamiltonian via the

Poisson bracket.

The significance of the Poisson bracket extends into many branches of math-

ematics and physics. In quantum mechanics, a Hamiltonian theory has the same

essential features as in classical mechanics except that the Poisson bracket goes

over to a simple commutator with the property that the position and momentum

operators do not commute,

[x, p] =

[x,−i~ ∂

∂x

]= i~.

A transformation to a new set of coordinates in phase space is said to be

canonical if it preserves the form of Hamilton’s equations. It is not necessary

to preserve the functional form of the Hamiltonian itself. This means that a

transformation of coordinates p → P and q → Q will result in a transformed

Hamiltonian H → K, but Hamilton’s equations stay the same with respect to

the new Hamiltonian K,

Q = {Q,K}, P = {P,K}.

Since a physicists goal is usually to find the essential conserved quantities or

47


symmetries for any system, it is convenient to find a canonical transformation

to new coordinates Q and P such that Q = P = 0. The easiest way to do this

is to canonically transform to a new Hamiltonian K such that K = 0, making

the equations of motion trivial but instantly revealing the relevant conserved

quantities as the new coordinates and their conjugate momenta.

Consider that any canonical transformation must leave the variation of the

action the same:

δS = δ

∫(pq −H)dt = δ

∫(PQ−K)dt = 0

although this does not mean the integrands themselves are exactly the same. The

variation of the action is unaltered up to the addition of any total time derivative

of a (at least) twice differentiable function of the generalised coordinates, F (q, t).

To see this, consider that

δS = δ

∫ b

a

(L(q, q, t) +

dF (q, t)

dt

)dt

=

∫ b

a

δLdt+∂F

∂qδq

∣∣∣∣ba

,

and the variation of q vanishes at the endpoints a and b, so the addition of dF/dt

to the action integrand has no effect on the variation of the action. Thus, a

canonically transformed action integrand must be related in general by

pq −H = PQ−K +dF (q,Q, t)

dt

(a scale transformation may also relate these expressions, but it is a trivial case).

This gives (p− ∂F

∂q

)q −H =

(P +

∂F

∂Q

)Q−K +

∂F

∂t

and since the coordinates q and Q are independent of each other, their coefficients

48


must vanish [42], ultimately yielding the equations

p =∂F

∂q, P = −∂F

∂Q, and K = H +

∂F

∂t.

Since a new Hamiltonian K = 0 is desirable to simplify the equations of

motion, this leaves

H +∂F

∂t= 0

and the function F takes on a very special form, since its total time derivative is

dF

dt=∂F

∂t+∂F

∂qq +

∂F

∂QQ

= pq −H (Q = 0 when K = 0),

which reveals F in this case to be the action itself (up to a constant).

Eqns (2.14) can be given by some other ‘noncanonical’ bracket and still indi-

cate a Hamiltonian theory, provided that the bracket is bilinear, antisymmetric

and satisfies the Jacobi identity [53],

{f, {g, h}}+ {h, {f, g}}+ {g, {h, f}} = 0.

Interestingly, a proof due to Darboux states that such a noncanonical bracket

can always be transformed back into the canonical Poisson bracket, at least locally

in some neighbourhood around a point in phase space [54].

This fact has been exploited by some [55, 56, 57, 57, 58, 59, 60, 61], includ-

ing this author [62], in deriving noncanonical Poisson brackets or noncanonical

coordinates to yield a Hamiltonian description of systems that were thought to

have no Hamiltonian structure. This is extremely useful, since many systems of

real physical interest are described in terms of variables that defy all attempts

to transform them into canonical coordinates. The Hamiltonian may however be

defined independent of a Lagrangian and in terms of coordinates that are not nec-

essarily canonical, provided the noncanonical Poisson bracket satisfies the three

49


properties mentioned above. For an excellent review on noncanonical methods

applied to fluids (and Hamiltonian mechanics in general), see [63].

Note that the definition of the Hamiltonian in Eqn (2.9) requires a concave

Lagrangian function and that Eqn (3.7) be invertible to q (otherwise H would

not be expressible in terms of just p, q). Dirac developed a more general theory,

known as Dirac constraint theory [5], that sidesteps these requirements. Given

a number of constraints φk(p, q) = 0 arising from problems with Eqns (2.9), or

merely from other constraints one might choose to impose on the system, the

equations of motion become

dqidt≈ {qi, H}+ uk{qi, φk},

where the symbol ‘≈’ represents weak equality, indicating that strong equality

follows by applying the constraints only after the Poisson brackets have been eval-

uated. The coefficients uk are determined by the requirement that the constraints

are constant in time;

dφidt≈ {φi, H}+ uk{φi, φk} ≈ 0.

Provided that the Lagrangian from which the Hamiltonian was derived is

consistent, there are only two relevant outcomes of evaluating the above equation.

Either a new secondary constraint is found (a function expressed just in terms

of the canonical variables), or a condition is found that the coefficient uk must

satisfy. If a secondary constraint is found, then this process must be repeated

again, and again for any further constraints found, until all that is left is a number

of constraints and conditions on their corresponding coefficients.

The total Hamiltonian HT is written as the ‘naive’ Hamiltonian H, con-

structed from a Legendre transformation of the Lagrangian, plus the constraints

50


φk(q, p) multiplied by their corresponding coefficients uk so that

HT = H + ukφk.

The equations of motion for the canonical coordinates are then correct with

respect to the time evolution generated by HT ;

dq

dt≈ {q,HT}

where the constraints are applied after evaluating the Poisson bracket. The use of

Hamiltonian mechanics in fluid and plasma physics will be reviewed in Chapter

3.

51

3

Lagrangian and Hamiltonian

Formulations of Laser-Plasma

Interaction

The focus of this thesis is on variational principles applied to laser-plasma inter-

action using either a fluid model or a kinetic model. This chapter will serve as

a review of all the relevant literature related to variational principles applied to

both fluid and kinetic models. While some of the literature does not deal specifi-

cally with plasmas, the underlying mathematical principles are the same for any

fluid or kinetic model, give or take certain assumptions that may be relevant

only to a plasma. All of this literature has a common thread in that they at-

tempt to describe the physics of fluids or gases using Lagrangian or Hamiltonian

formalisms.

Variational principles in fluid dynamics were used as early as 1929 by Bateman

[64], but the truly influential papers on the subject were due to the likes of Taub

[65, 66], Davydov [67], Herivel [68] and Eckart [69].

52

3.1 The Boltzmann-Vlasov Distribution

3.1 The Boltzmann-Vlasov Distribution

A canonical description specifically tailored for plasmas was first considered by

Low [44] who sought to generalise Taub’s hydrodynamic method [65] to a plasma

described by a Boltzmann-Vlasov equation. Low considered an initial Boltzmann-

Vlasov distribution (at time t = 0) given by f(x0,v0) and let the particle position

at any time t, x(x0,v0, t), be a solution to the equation of motion that returns

the initial coordinates and velocities when t = 0 such that

x(x0,v0, 0) = x0 and v(x0,v0, 0) = v0.

Under a change of coordinates from x → x0 and v → v0, which has unit

Jacobian, the Boltzmann-Vlasov distribution can be transformed such that

f(x,v, t)→ f(x0,v0).

The total plasma Lagrangian L is then given by the integral of a Lagrangian

density L for matter and interaction with respect to the distribution function

over all of phase space, plus the total electromagnetic energy;

L =

∫ ∫f(x0,v0)Ldx0dv0 +

∫ (ε02E2 − 1

2µ0

B2

)dx.

If the Lagrangian density is the standard one for a particle in an electromag-

netic field,

L =1

2m

(∂x

∂t

)2

− qφ+ q∂x

∂t·A,

then Low found that Hamilton’s Principle,

δS =

∫δLdt = 0,

where the quantities x, φ and A are independently varied yields the Lorentz force

53

3.2 Ideal Fluids

and Maxwell’s inhomogeneous equations:

m∂2x

∂t2= q(E + v ×B);

∇ · E =q

ε0

∫f(x,v, t)dv;

∇×B = µ0q

∫f(x,v, t)dv +

1

c2∂E

∂t.

3.2 Ideal Fluids

An ideal fluid description using Hamilton’s principle was given by Herivel [68]

who was unsatisfied by the earlier efforts of Lichtenstein [70] and Taub [65].

Lichtenstein had assumed no variation in temperature in applying Hamilton’s

principle, while Taub had defined temperature as the time derivative of another

(physically meaningless) function and then applied the constraint of constant

entropy after the application of Hamilton’s principle. These approaches seemed

ad hoc to Herivel, who considered the true power of Hamilton’s principle to lie

in its ability to derive equations of motion, rather than assume them, in the

process of describing a system. Herivel proposed that moving from a discrete to

continuous (fluid) description requires changing the form of the Lagrangian by

subtracting an additional term, the internal energy U(n, S) of the fluid, so that

L = T − V − U where T is the kinetic energy of the fluid and V the potential

energy.

In §1.3.1, two equivalent formulations of fluid mechanics were reviewed using

either Eulerian or Lagrangian coordinates. Herivel considered both cases and

used essentially the same approach that Low [44] later applied to the Maxwell-

Boltzmann description of a plasma by transforming the volume integral in the

action into one depending on the initial Lagrangian coordinates a, rather than

54

3.2 Ideal Fluids

the usual Lagrangian coordinates ξ(a, t) so that

S =

∫ ∫ (1

2nξ2 − n(V + U)

)Jdadt

where

J =∂ξ

∂a.

In this way, the action integral uses a volume element comoving with the fluid,

and all that remains is to apply the constraint

n(ξ, t) =n(a)

J(3.1)

(a form of the continuity equation in Lagrangian coordinates) and the isentropic

constraint (see §1.3.3). In Lagrangian coordinates, constant entropy is expressed

as (∂S

∂t

)a

= 0. (3.2)

The above method of transforming coordinates and functions is ubiquitous

in the literature discussing fluid mechanics using Lagrangian coordinates [see

61, 68, 69, 71, 72, 73, 74]. Both (3.1) and (3.2) can be added to the action

integral using Lagrange multipliers α and β such that

S =

∫ ∫ (1

2nξ2 − n(V + U)− α(n(ξ, t)J − n(a))− β

(∂S

∂t

)a

)dadt.

Applying Hamilton’s principle where ξ, n and S are independently varied

(variation with respect to α and β simply returns the constraints), Herivel ulti-

mately found the set of equations:

δn : α =1

2ξ2 − V − U − P

n

δS :∂β

∂t= nT

δξ :∂2ξ

∂t2= −∇V − 1

n∇P

55

3.2 Ideal Fluids

The last equation found by varying ξ is a form of Euler’s equation (1.21) that

includes the gradient of any other potential field that may affect the fluid (gravity,

for instance). The same basic procedure was used relatively recently by Antoniou

& Pronko [74] to give a Hamiltonian description of a plasma.

Herivel achieved the same results in Eulerian coordinates. In this case, the

volume integral in the action is with respect to the Eulerian coordinates x and

the velocity is the Eulerian velocity v(x, t). The functions n and S are also now

expressed in Eulerian coordinates. As such, the constraints must be modified for

the new coordinates, where the continuity equation is the usual

∂n

∂t+∇ · (nv) = 0

and constant entropy flux density is given by

ndS

dt= 0

where d/dt is now of course the material derivative. Inserting these constraints

into the action using Lagrange multipliers α and β gives

S =

∫ ∫ (1

2nv2 − n(V + U)− α

(∂n

∂t+∇ · (nv)

)− βndS

dt

)dxdt.

In the Lagrangian case, the third variable apart from n and S that was to be

independently varied was the fluid coordinate ξ. Now in the Eulerian case, the

velocity will be varied independently in applying Hamilton’s principle and the set

of equations Herivel found from these variations (after some working) is

δn :dv

dt= −∇V − 1

n∇P

δS :∂β

∂t+ v · ∇β = 0

δv : v = −∇α + β∇S

56

3.3 Clebsch Potential Representation

Euler’s equation has again been recovered by an application of Hamilton’s

principle, but this time with respect to the Eulerian coordinates. Herivel noticed

that the final equation from δv represents a decomposition of the vector field

with respect to three scalar functions, which is a special case of something called

the Clebsch potential representation that will be discussed in §3.3). This was the

weakness of Herivel’s Eulerian approach; the velocity is expressed only in terms

of ‘unphysical’ potentials and is not in fact fully general [71]. An alternative to

the above method of varying the Eulerian velocity will be discussed in §3.4.


Eckart [69] generalised Herivel’s approach to the case of a compressible fluid, but

it was Lin [75] who noticed that an additional constraint introduced to the action

would make Herivel’s Eulerian velocity expression fully general. The additional

term was of the form

ndλidt,

introduced as a constraint to the action integral with the Lagrange multiplier γi.

With this extra constraint, the variation with respect to the Eulerian velocity

yielded the expression

v = −∇α + β∇S + γi∇λi.

The velocity was now a fully general expression at the cost of introducing six

new scalar functions with no intuitive physical interpretation. Seliger & Whitham

[76] showed that Lin’s generalisation could be simplified by pointing out the

connection with a result of Clebsch [77]. Clebsch had shown that any vector field

could be generally decomposed into an expression in terms of only three scalar

functions such that

v = α∇β +∇ψ. (3.3)

Chen & Sudan [78] used the Clebsch potentials specifically in their research

57


related to laser-plasma interaction. They considered the canonical relativistic

momentum in an electromagnetic field given by

P = mvγ − qA = α∇β +∇ψ. (3.4)

For the special case of a curl free vector field (∇ × P = 0), the decompo-

sition is simply P = ∇ψ since in general ∇α × ∇β 6= 0. Chen & Sudan were

interested in cold relativistic plasmas. The equations governing cold, relativistic

laser-plasma fluid interactions are Maxwell’s two inhomogeneous equations (as

usual, the homogeneous equations follow from the definition of E and B in terms

of the potentials), the Lorentz force and the continuity equation:

∇×B = −µ0J +1

c2∂E

∂t;

∂p

∂t+ v · ∇p = q(E + v ×B);

∂n

∂t= −∇ · nv.

Note that the Lorentz force was given in terms of the material derivative of

the momentum p, which must be so in the case of fluid dynamics.

The relativistic Lagrangian for laser-plasma interaction selected by Chen &

Sudan was

L = nmc2(1− γ−1) + qnφ− qnv ·A− 1

2µ0

B2 +ε02E2. (3.5)

Given the vector calculus identity,

∇(A ·B) = A · ∇B + B · ∇A + A× (∇×B) + B× (∇×A), (3.6)

58


the Lorentz force can be expressed in terms of the canonical momentum:

∂P

∂t= −v · ∇mvγ +∇qφ− qv × (∇×A)

= −v · ∇mvγ +∇qφ− v × (∇× (mvγ −P))

= ∇(qφ−mv2γ) +mvγ · ∇v + v × (∇×P) (using 3.6)

= ∇(qφ−mv2γ −mc2γ−1) + v × (∇×P)

= ∇(qφ−mc2γ) + v × (∇×P).

(3.7)

But now, given the ‘Lin constraints’ [75]

dα(x, t)

dt=dβ(x, t)

dt= 0, (3.8)

the Lorentz force can also be expressed in terms of Clebsch potentials:

∂(α∇β +∇ψ)

∂t= ∇(qφ−mc2γ) + v × (∇α×∇β)

∇(α∂β

∂t+∂ψ

∂t

)−∇α∂β

∂t+∇β∂α

∂t= ∇(qφ−mc2γ) + (v · ∇β)∇α− (v · ∇α)∇β

=⇒ α∂β

∂t+∂ψ

∂t= qφ−mc2γ + C (using 3.8)

where C is a constant determined by initial conditions. Since the plasma may be

considered static prior to the arrival of the laser pulse, at time t = 0, γ = 1 and

∂ψ/∂t = α = β = 0 which gives C = mc2 and the Lorentz force is

α∂β

∂t+∂ψ

∂t= qφ−mc2(γ − 1). (3.9)

The Lagrangian itself (3.5) can also be expressed independently of v assuming

59


that momentum is conserved, i.e., dψ/dt = 0, given that

mc2(1− γ−1)− qv ·A = mc2 +mv2γ −mc2γ − qv ·A

= v · (mvγ − qA)−mc2(γ − 1)

= v · (α∇β +∇ψ)−mc2(γ − 1)

= −α∂β∂t− ∂ψ

∂t−mc2(γ − 1).

Therefore, the Lagrangian 3.5 can be expressed in terms of the Clebsch po-

tentials and electromagnetic potentials as

L = −n(α∂β

∂t+∂ψ

∂t+mc2(γ − 1)− qφ

)− 1

2µ0

B2 +ε02E2. (3.10)

Note that the relativistic factor γ can also be expressed independent of v

given that

γ =1√

1− v2/c2=⇒ γ2v2 = c2(γ2 − 1)

and

P = mvγ − qA =⇒ γ2v2 =(P + qA)2

m2,

which gives

γ =√

1 + (P + qA)2/m2c2.

However, Chen & Sudan argued that prior to a laser pulse hitting the plasma,

the plasma electrons have zero canonical momentum and so the curl of the mo-

mentum is also zero. However, taking the curl of the Eqn (3.7) shows that this

will be true for all time and so the canonical momentum can be considered to be

curl-free in this scenario and expressible in terms of just one potential, ψ. In this

case, the Lagrangian density is just

L = −n(∂ψ

∂t+mc2(γ − 1)− qφ

)− 1

2µ0

B2 +ε02E2 (3.11)

60


and the Lorentz factor is

γ =√

1 + (∇ψ + qA)2/m2c2.

Now Hamilton’s Principle can be applied to the action

S =

∫Ld3xdt (3.12)

where L is given by 3.11 and the generalised coordinates to be independently

varied are ψ,A, φ, n. However, even though α and β no longer appear in the

action integral, it is worth noting what their variation would yield in the general

case where L is (3.10). Since there are no derivatives of α appearing in 3.10,

independently varying α simply yields the equation

∂L

∂α=∂β

∂t+mc2

∂γ

∂α= 0. (3.13)

The relativistic factor γ is a function of P = α∇β +∇ψ and A, so

∂γ

∂α=

∂

∂α

(1 +

α2(∇β)2 + 2α∇β · ∇ψ + (∇ψ)2 + 2q(α∇β ·A +∇ψ ·A) + q2A2

m2c2

)1/2

=α(∇β)2 +∇β · ∇ψ + q∇β ·A

m2c2γ

= ∇β · mvγ

m2c2γ

= ∇β · v

mc2

Therefore, equation 3.13 returns one of the Lin constraints (the constraint

that β is ‘carried’ along with the fluid flow):

∂β

∂t+mc2

∂γ

∂α=∂β

∂t+ v · ∇β =

dβ

dt= 0.

The reappearance of the velocity in the calculations above is merely a conve-

nience to transit to the final expression.

61


There are derivatives of β appearing in 3.10, so varying β yields the equation

∂L

∂β− ∂

∂t

∂L

∂β− ∂

∂xi∂L

∂ ∂β∂xi

= 0, (3.14)

which gives

∂α

∂t+ v · ∇α =

dα

dt= 0

since

∂γ

∂∇β=

α

mc2v.

Now to the heart of the matter, for the case where α = β = 0, varying ψ in

(3.12) gives the equation

∂L

∂ψ− ∂

∂t

∂L

∂ψ− ∂

∂xi∂L

∂ ∂ψ∂xi

= 0. (3.15)

Note that the derivatives in equation 3.15 are partial derivatives as it is these

partial derivatives of ψ that are treated as independent variables in the integration

by parts that takes place when finding the action extremal. The distinction

is emphasised here especially to point out that ψ - the time derivative of the

generalised coordinate ψ - is not treated as the material derivative d/dt = ∂/∂t+

v · ∇ for the purposes of applying Hamilton’s principle. Now equation 3.15 gives

the continuity equation,

∂n

∂t+∇ · ( n

mγ(∇ψ + qA)) = 0, (3.16)

Therefore, it has been shown that independently varying α and β in Equation

3.12 and demanding that δS = 0 yields the Lin constraints for β and α respec-

tively, while varying ψ yielded the equation of continuity. This process can be

repeated for the remaining generalised coordinates φ and A. To summarise, the

coordinate variations are matched below with the equation yielded by Hamilton’s

62

3.4 Virtual Fluid Displacement

Principle:

δα :dα

dt= 0,

δβ :dβ

dt= 0,

δψ :∂n

∂t+∇ · ( n

mγ(∇ψ + qA)) = 0,

δφ : ∇ · E =qn

ε0,

δA : ∇×B = µ0qn

mγ(∇ψ + qA) +

1

c2∂E

∂t,

δn :∂ψ

∂t= qφ−mc2(γ − 1).

Chen & Sudan therefore succeeded in applying Hamilton’s principle to the

case of cold, relativistic laser-plasma interaction where the plasma is treated as

a fluid with Eulerian velocity v. Some authors find the decomposition of v in

terms of the Clebsch potentials to be useful as a mathematical tool, but find their

‘unphysicality’ to be a barrier to deeper understanding of a given problem.


While Chen & Sudan [78] and Seliger & Whitham [76] considered the Clebsch

potential representation to solve the difficulties of Hamilton’s Principle applied to

a fluid system using Eulerian coordinates, Newcomb had earlier taken a different

approach [79] to a variational formulation of magnetohydrodynamics applied to

the case of an infinitely conductive, non-relativistic plasma. The crux of the

problem was still the difficulties in applying a variational principle to a fluid

theory in terms of Eulerian coordinates (this was the reason that other authors

explored Clebsch potential formulations). Newcomb was able to reproduce certain

magnetohydrodynamic equations in both Eulerian and Lagrangian coordinates

(see §1.3.1) via Hamilton’s principle and demonstrated. Herivel expressed the

pressure P , density n and magnetic field B in Lagrangian coordinates with respect

63


to the initial configuration of the system in the usual way (see §3.2)) such that

n =n(a)

J,

P =P (a)

Jκ,

Bi =∂ξi∂aj

Bi(a)

J,

where

J = det

(∂ξ

∂a

)and κ is the adiabatic index (κ = CP/CV - ratio of specific heat at constant

pressure over constant volume). The set of magnetohydrodynamic equations that

Newcomb sought to reproduce (expressed here all in Eulerian coordinates) was

∂n

∂t+∇ · (nv) = 0,

∂P

∂t+ v · ∇P + κP∇ · v = 0,

∂B

∂t−∇× (v ×B) = 0,

∇ ·B = 0,

n

(∂v

∂t+ (v · ∇)v +∇φ

)+∇

(P +

1

2B2

)−B · ∇B = 0.

(3.17)

The first equation above is the usual continuity equation. The second equa-

tion is the adiabatic ideal gas law and the third equation is a combination of

Faraday’s Law with the Lorentz force law where infinite conductivity is assumed

(−E = v × B). The last equation is another way of expressing the Lorentz

force law where the time varying electric field has been neglected in Ampere’s

law in calculating J × B = (∇ × B) × B. This particular set of equations is

non-relativistic and also uses a scalar pressure P , despite the assumption of in-

finite conductivity (usually only assumed for collisionless plasmas with tensorial

64


pressure). Newcomb argued that these assumptions are justified in certain as-

trophysical regimes. The particular form of these equations is not particularly

relevant to this thesis, indeed, laser-plasma interaction was not Newcomb’s in-

terest in studying this system of equations, but his work has influenced other

authors who used similar ideas in direct application to laser-plasma physics. It is

the particular way in which Newcomb solved a problem in fluid mechanics using

Hamilton’s principle that is relevant here.

Newcomb guessed the form of the Lagrangian for the system of equations in

(3.17) based on the standard expression as a difference of the kinetic and potential

energy. In Lagrangian coordinates Newcomb’s Lagrangian density was

L = n

(1

2ξ2 − ϕ

)− P

(κ− 1)Jκ−1− 1

2µ0J

∂ξi∂aj

∂ξi

∂akBjBk (3.18)

where ϕ is the gravitational potential energy. Variation of the action dependent

on this Lagrangian density with respect to the Lagrangian coordinates ξ yields

the system of equations in (3.17) once converted back to Eulerian coordinates,

proving that Newcomb’s choice of Lagrangian was the correct one.

The most important ingredient in Newcomb’s work (for the purposes of this

thesis) was his derivation of the variation of the Eulerian fluid velocity, which is of

course necessary for any variational theory attempting to deal with Eulerian fluid

dynamics without the use of Clebsch potentials. Newcomb used the definition of

Eulerian velocity in terms of Lagrangian velocity,

v(x, t)|x=ξ(a,t) = ξ(a, t), (3.19)

together with a definition of virtual fluid displacement z, (variation of Eulerian

fluid coordinates in terms of Lagrangian particle coordinates)

z(x, t)|x=ξ(a,t) = δξ(a, t) (3.20)

65


to define a variation in the Eulerian fluid velocity. The variation in the fluid

velocity will be considered while following a particular fluid particle so that the

variation of Eqn (3.19) is done while holding a, the initial position of a Lagrangian

particle, constant so that

δv|x=ξ(a,t) = δξ − (δx · ∇)ξ

=⇒ δv + (z · ∇)v = δξ.

(3.21)

Next, simply take the time derivative of Eqn (3.20) to get

dz

dt

∣∣∣∣x=ξ(a,t)

=∂z

∂t+ (v · ∇)z = δξ. (3.22)

Combining Eqns (3.21) and (3.22) gives

δv =∂z

∂t+ (v · ∇)z− (z · ∇)v, (3.23)

which is the variation of the Eulerian fluid velocity in terms of the virtual fluid

displacement z. The variations in density, scalar pressure and magnetic field can

also be defined in terms of the virtual fluid displacement where

δn = −∇ · (nδξ), (3.24)

δP = −κP (∇ · z)− z · ∇P, (3.25)

and

δB = ∇× (z×B). (3.26)

The Lagrangian from (3.18), now expressed in Eulerian coordinates is

66


L = n

(1

2v2 − ϕ

)− P

κ− 1− 1

2µ0

B2.

It was with this expression that Newcomb applied Hamilton’s principle using

Eqns (3.23), (3.24), (3.25) and (3.26) to yield the system of equations set out in

(3.17), showing that his method of variation with respect to the Eulerian virtual

fluid displacement was also correct.

Newcomb’s method of varying the Eulerian fluid velocity was used by Brizard

[80] [81] to specifically study the case of laser-plasma interactions for a cold rel-

ativistic plasma. The Lagrangian considered by Brizard was

L = nmc2(1− γ−1)− q(ne − ni)φ+ qnv ·A− 1

2µ0

B2 +ε02E2, (3.27)

which is simply a relativistic Lagrangian (see §2.4) for two charged fluids, one

corresponding to an electron fluid with density ne and the other to an ion fluid

with density ni. Brizard used Newcomb’s expression for the variation of the

Eulerian fluid velocity and density - Eqns (3.23) and (3.24) - in terms of the

virtual fluid displacement in applying Hamilton’s principle to (3.27).

The variation of (3.27) is therefore given by

δL =∂L

∂nδn+

∂L

∂v· δv +

∂L

∂φδφ+

∂L

∂A· δA +

∂L

∂B· δB +

∂L

∂E· δE

= −∂L∂n∇ · (nδξ) +

∂L

∂v·(∂δξ

∂t+ v · ∇δξ − δξ · ∇v

)+∂L

∂φδφ+

∂L

∂A· δA

+∂L

∂B· (∇× δA) +

∂L

∂E·(−∇δφ− ∂δA

∂t

)= δφ

(∂L

∂φ+∇ · ∂L

∂E

)+ δA ·

(∂L

∂A+∂

∂t

(∂L

∂E

)+∇× ∂L

∂B

)− δξ ·

(∂

∂t

(∂L

∂v

)− n∇∂L

∂n+∇ ·

(∂L

∂v⊗ v

)+∂L

∂v· ∇v

)+∂

∂t

(∂L

∂v· δξ − ∂L

∂E· δA

)+∇ ·

(∂L

∂B× δA− ∂L

∂nnδξ +

∂L

∂v· (v ⊗ δξ)− ∂L

∂Eδφ

)

67


Note that the varied Lagrangian is now split into terms multiplied by the arbi-

trary variations of φ,A, ξ, a partial time derivative and a partial space derivatives.

Hamilton’s principle yields the set of equations,

δφ : ∇ · E =qn

ε0,

δA : ∇×B = µ0qnv +1

c2∂E

∂t,

δξ :∂p

∂t+ v · ∇p = q(E + v ×B).

The remaining part of the varied Lagrangian gives the energy conservation

and momentum conservation equations upon application of Noether’s theorem.

For the equations derived above, the field variables were varied independently,

that is their functional form was varied while holding their spacetime dependence

constant. Consider now time translations of the variables φ and A, where the

functional form is held constant while the time dependence is varied:

δA =∂A

∂tδt, δφ =

∂φ

∂tδt and δξ = vδt.

The equation regarding conservation of energy is then given by

∂

∂t

(∂L

∂v· v − ∂L

∂E· ∂A

∂t

)(3.28)

+∇ ·(∂L

∂B× ∂A

∂t− n∂L

∂n⊗ v +

∂L

∂v· (v ⊗ v)− ∂L

∂E

∂φ

∂t

)δt = 0

which is the same law of conservation of energy derived in §2.3, except where a

current is present;

1

2

∂

∂t

(ε0E

2 +1

µ0

B2

)+∇ · S = −qnv · E.

The law of conservation of momentum is found by considering spatial trans-

lations of the variables φ and A where

68

3.5 Relativistic Constraints in Fluid Dynamics

δφ = (δx · ∇)φ, δA = (δx · ∇)A and δξ = δx,

so that Hamilton’s principle gives

∂

∂t

(∂L

∂v· v − ∂L

∂E· ∇A

)δx (3.29)

+∇ ·(∂L

∂B×∇A− ∂L

∂nn⊗ v +

∂L

∂v· (v ⊗ v)− ∂L

∂E∇φ)δx = 0

which is

∇ ·U− 1

c2∂S

∂t=dnv

dt.

The equations governing relativistic cold laser-plasma interaction can there-

fore be derived using Eulerian coordinates. The Clebsch potential formalism

relies on the ‘unphysical’ potential functions α, β and ψ and the emergence of

the equations from a variational principle is not intuitive. In contrast to this, the

Eulerian formulation elaborated upon by Brizard gives the Lorentz force law from

the variation of the virtual fluid displacement analogous to the way Newton’s law

of motion can be found from varying the physical coordinates of a particle in

classical mechanics.

Brizard et al. [82] also used this general method of variation in considering

neutrino-plasma interactions using a fluid model, although a discussion of neutri-

nos is beyond the scope of this thesis. The subject of nonlinear plasma dynamics,

treated as a fluid, was also explored in detail by Zakharov et al. [61, 83] using a

Hamiltonian theory that did not rely on a Lagrangian description to begin with.


In special relativity, the proper time and four-velocity are no longer independent

of each other but are constrained by the relationship

dξαdξα = c2dτ 2.

69


Therefore, a variation of the infinitesimal proper time is

δdτ =1

cδ(√dξαdξα) =

1

c2dξαdτ

dδξα (3.30)

which implies a variation in the four-velocity

δ

(dξµdτ

)=dδξµdτ− dξµ

(dτ)2δdτ =

dδξµdτ− 1

c2dξµdτ

dξαdτ

dδξα

dτ. (3.31)

These constraints were addressed long ago by Infeld [84] and Kalman [85].

Cavalleri and Spinelli [86, 87, 88] attempted to take into account curvilinear co-

ordinates in General Relativity or relativistic compressible fluids using the above

variation, research that was repeated by Brown [89] many years later, perhaps

more rigorously. Kalman noted [85] that, given the constraint 3.31 (which also

implies uαuα = c2), the variation of the action was

δS =

∫(δL)dτ +

∫L(δdτ)

=

∫ (∂L

∂ξαδξα +

∂L

∂uαδuα)dτ +

1

c2

∫Ldξαdτ

dδξα

=

∫ (∂L

∂ξαδξα +

∂L

∂uαdδξαdτ− 1

c2∂L

∂uαdξαdτ

dξβdτ

dδξβ

dτ

)dτ +

1

c2

∫Ldδξα

dτ

dξαdτ

dτ

=

∫ (∂L

∂ξα− d

dτ

(∂L

∂uα− 1

c2∂L

∂uβuβuα +

1

c2Luα

))δξαdτ

where uµ = dξµ/dτ . The equality between the third and fourth lines above is

given by integration by parts, again using the fact that the surface terms must

vanish due to the vanishing variations at the endpoints. Therefore, by demanding

δS = 0 and given that this must be satisfied for any arbitrary variations δξ, the

correct relativistic equations of motion are

d

dτ

(∂L

∂uα− 1

c2uα

(∂L

∂uβuβ − L

))=

∂L

∂ξα. (3.32)

70


Cavalleri and Spinelli first generalised Kalman’s result for a test particle in

general relativity [86] and then explored variations of an element of a perfect fluid

in special relativity [87]. While general relativistic theories are beyond the scope

of this thesis, the application of constrained variations to a special relativistic

perfect fluid will be reviewed here.

The total Lagrangian L is an integral of the Lagrangian density L over a

3-dimensional volume,

L =

∫LdV ≈ L∆V,

where ∆V is a fluid element (allowed to be compressible). Now the variation of

the action is

δS = δ

∫L∆V dτ =

∫(δL)∆V dτ +

∫L∆V (δdτ) +

∫L(δ∆V )dτ. (3.33)

The variations of L are in the same form as those for L, and the variation

of dτ was given in Eqn (3.30). Now it remains to find the δ∆V . Cavalleri &

Spinelli considered that the change in pressure P caused by varying ∆V must be

balanced by a change in the internal energy such that

Pδ(∆V ) = c2δ(ρ0∆V ) = c2(δρ0∆V + ρ0δ(∆V )) (3.34)

where ρ0 is the proper mass density. Rearranging terms in Eqn (3.34) gives

δ(∆V ) = − c2∆V

c2ρ0 + P

∂ρ0∂xα

δxα (3.35)

where the variation in mass density is

δρ0 =∂ρ0∂xα

δxα. (3.36)

A variation in the fluid volume element has now been taken into consideration

in varying the action and again demanding that δS = 0 gives the new equations

71


of motion

d

dτ

(∂L

∂uα+

1

c2uα

(L− ∂L

∂uβuβ

))=

∂L

∂xα−(∂L

∂uα+

1

c2uα

(L− ∂L

∂uβuβ

))u ,γγ

− c2L

c2ρ0 + p

∂ρ0∂xα

(3.37)

As expected, when the mass in the volume element ∆V remains constant, Eqn

3.37 reduces to Eqn 3.32. Cavalleri and Spinelli considered a Lagrangian density

of the form

L = c2ρ0gαβuαuβ + Pgαβu

αuβ

and substituting this into Eqn (3.37), using the continuity equation

(ρ0 +

P

c2

)uβ,β = −dρ0

dτ,

gives the equation of motion as

c

(ρ0 +

P

c2

)duαdτ

+1

c

dP

dτuα =

∂P

∂xα. (3.38)

However, while Cavalleri & Spinelli were discussing a fluid, the distinction be-

tween Eulerian and Lagrangian coordinates was not made clear in [87]. This was

compounded by the fact that Kalman’s original paper [85] and the generalisation

by Cavalleri & Spinelli themselves [86] both dealt with the Lagrangian coordinates

of a single test particle. It is unclear in Eqn (3.37) whether the four-velocity repre-

sents the Eulerian fluid velocity or the Lagrangian fluid particle velocity. Indeed,

the spacetime coordinates considered in the variation of the four-velocity and

proper time appear to be the same as those considered in the variation of the

mass density (3.36), which would imply an Eulerian four-velocity. However, as

was seen in §3.4, the variation of an Eulerian velocity is not so straight-forward,

so this author questions whether the variational principle resulting in (3.32) can

be applied without alteration to the case of a perfect fluid.

72


This author presents the following argument for a variation of the Eulerian

four-velocity in line with Newcomb’s method [79] for the three-dimensional non-

relativistic case. Consider the relativistic generalisation of Eqns (3.19) and (3.20)

such that the Eulerian four-velocity is related to the Lagrangian four-velocity

whereby

uα(xµ, τ)|xν=ξν(aω ,τ) =dξα(aµ, τ)

dτ(3.39)

and the virtual fluid displacement is

zα(xµ, τ)|xν=ξν(aω ,τ) = δξα(aµ, τ). (3.40)

The variation of Eqn (3.39) while following one fluid particle (that is, elimi-

nating the dependence of dξ/dτ on initial spacetime coordinate) is

δuα|xν=ξν(aω ,τ) = δ

(dξαdτ

)− δxβ ∂

∂xβ

(dξαdτ

),

This gives

δuα + zβ∂uα∂xβ

= δ

(dξαdτ

). (3.41)

Now taking the derivative of Eqn (3.40) with respect to proper time and

recalling Eqn (3.31,

dzαdτ

=dδξαdτ

= δ

(dξαdτ

)+

1

c2uαuβ

dδξβ

dτ. (3.42)

Substituting Eqns (3.40) and (3.42) into Eqn (3.41) gives the variation of the

Eulerian four-velocity for a fluid in terms of a virtual fluid displacement:

δuα =dzαdτ− 1

c2uαuβ

dzβ

dτ− zβ ∂uα

∂xβ. (3.43)

This differs from Kalman’s expression for the varied four-velocity by the last

term which takes into account the dependence of an Eulerian velocity on position

in space. Eqn (3.43) is however in agreement with another more general expres-

73


sion derived by Achterberg [90] which took into account curvilinear coordinates.

Now by substituting Eqns (3.30), (3.35) and (3.43) into Eqn (3.33), Hamilton’s

principle becomes, in terms of virtual fluid displacement

δS =

∫ (∂L

∂xαzα +

∂L

∂uα

(dzα

dτ− 1

c2uαuβ

dzβ

dτ− zβ ∂u

α

∂xβ

))∆V dτ

+

∫1

c2Luα

dzα

dτ∆V dτ −

∫c2L

c2ρ0 + P

∂ρ0∂xα

zα∆V dτ

Integrating by parts, and given that the surface terms vanish due to the virtual

fluid displacement zα disappearing at the endpoints, this becomes

δS =

∫ (∂L

∂xα− ∂L

∂uβ∂uβ

∂xα− 1

∆V

d

dτ

((∂L

∂uα− 1

c2uαu

β ∂L

∂uβ

)∆V

))zα∆V dτ

−∫ (

1

∆V

d

dτ

(1

c2Luα∆V

)− c2L

c2ρ0 + P

∂ρ0∂xα

)zα∆V dτ

Note that the additional term in Eqn (3.43) will now cancel with the explicit

derivative of L with respect to the Eulerian spacetime coordinates. Also, it is

true that

1

∆V

d∆V

dτ= uβ,β.

The requirement that δS = 0 and the arbitrariness of the virtual fluid dis-

placement finally gives the equation of motion

d

dτ

(∂L

∂uα− 1

c2uαu

β ∂L

∂uβ+

1

c2uαL

)= −

(∂L

∂uα− 1

c2uαu

β ∂L

∂uβ+

1

c2uαL

)uβ,β

− c2L

c2ρ0 + P

∂ρ0∂xα

,

which differs from Cavlleri & Spinelli’s original expression, (3.37), only in that the

term ∂L/∂xα is gone. In the equation of motion (3.38), this difference amounts

74

3.6 Maxwell-Vlasov System

to replacing the partial derivative of the pressure with a partial derivative of the

mass density such that

c

(ρ0 +

P

c2

)duαdτ

+1

c

dP

dτuα = −c2 ∂ρ0

∂xα. (3.44)

Relativistic laser-plasma interaction has also been investigated by Evstatiev

et al. [48] who used a Lagrangian formulation similar to that used by Brizard

[80] to formulate a relativistic Hamiltonian theory.


A Maxwell-Vlasov system is described by three equations - Ampere’s Law and

Farday’s Law coupled with the Vlasov equation:

1

c2∂E

∂t= ∇×B− µ0qα

∫vfα(z, t)δ(z− z0)dz,

∂B

∂t= −∇× E,

∂fα∂t

= −v · ∂fα∂x− eαmα

∫(E + v ×B)

∂fα∂v

δ(z− z0)dz.

(3.45)

The index α identifies the charge, mass and distribution function for different

species of particles in the plasma. The last equation is the Vlasov equation, which

is simply Liouville’s theorem in disguised form. Given some particle distribution

function f(x,v, t), Liouville’s theorem states that the distribution function is

constant along its trajectory in phase space;

df

dt= v · ∂f

∂x+ a · ∂f

∂v+∂f

∂t= 0.

It is clear that the acceleration in the above equation can be replaced by the

Lorentz force expression divided by mass to give the Vlasov equation.

Morrison, Weinstein and Marsden showed that the Maxwell-Vlasov system of

75


equations can be cast into the form of a continuous Hamiltonian system [55, 91,

92].

Given the Hamiltonian

H =

∫1

2mαv

2fαdz +

∫1

2(E2 +B2)dx

and a bilinear, anticommutative bracket that satisfies the Jacobi identity,

[F,G] =

∫f

{δF

δf,δG

δf

}dxdv +

∫ (δF

δE· ∇ × δG

δB− δG

δE· ∇ × δF

δB

)dx

+

∫δF

δE· ∂f∂v

δG

δE− δG

δE· ∂f∂V

δF

δfdxdv +

∫fB ·

(∂

∂v

δF

δf× ∂

∂v

δG

δf

)dxdv

where {F,G} is the canonical Poisson bracket given in Eqn (2.12), Eqns (3.45)

can be cast in Hamiltonian form so that

∂Bi

∂t= [Bi, H],

∂Ei∂t

= [Ei, H],∂fα∂t

= [fα, H].

3.6.1 Magnetohydrodynamics

A more general noncanonical Hamiltonian formalism was found for hydrodynam-

ics and magnetohydrodynamics (for an ideal fluid) by Morrison and Greene [57]

[56] in terms of the physical variables n, v, B and S (respectively fluid density,

velocity, magnetic field and entropy). The equations to be cast in Hamiltonian

form are

∂v

∂t= −∇v

2

2+ v × (∇× v)− 1

n∇(n2∂U

∂n

)+

1

n(∇×B)×B,

∂n

∂t= −∇ · (nv),

∂B

∂t= ∇× (v ×B),

∂S

∂t= −v · ∇S

(3.46)

76


where U(n, S) is the internal energy per unit mass. This is similar to the set of

equations originally considered by Herivel [68] in §3.2, except that they are now

for the case of magnetohydrodynamics, that is, electrically conducting fluids.

Morrison and Greene directly took the Hamiltonian as corresponding to the

total energy density of the fluid,

H =1

2nv2 + nU +

1

2B2,

bypassing the need for a Lagrangian. The disadvantage to this approach is that

none of the variables are canonical in the sense that the usual form of Hamilton’s

equations will not give their time evolution. However, by defining a noncanonical

Poisson bracket (see §2.5),

[F,G] = −∫V

δF

δn∇ · δG

δv+δF

δv· ∇δG

δn+δF

δv·(∇× v

n× δG

δv

)(3.47)

+1

n∇S ·

(δF

δS

δG

δv− δG

δS

δF

δv

)+

1

n

δF

δv·(

B×(∇× δG

δB

))+δF

δB·(∇×

(B× 1

n

δG

δv

))dτ,

Eqns (3.46) take the familiar form

∂vi∂t

= [vi, H],∂Bi

∂t= [Bi, H],

∂n

∂t= [n,H],

∂S

∂t= [S,H]. (3.48)

All these equations can be transformed to Eulerian variables (usually preferred

for practical considerations, especially as in this case the fluid density appears in

the denominator of some terms in Eqn (3.47)) n, B, σ = nS and M = nv where σ

is the specific entropy and M is the momentum density. The transformed Poisson

77

3.7 Guiding Centre Motion

bracket is then

[F,G] = −∫V

n

(δF

δM· ∇δG

δn− δG

δM· ∇δF

δn

)+ M ·

(δF

δM· ∇ δG

δM− δG

δM· ∇ δF

δM

)+ σ

(δF

δM· ∇δG

δσ− δG

δM· ∇δF

δσ

)+ B ·

(δF

δM· ∇δG

δB− δG

δM· ∇δF

δB+

(∇δFδB

)· δGδM−(∇δGδB

)· δFδM

)dτ

The equations of motion are then in the same Hamiltonian form as in Eqns

(3.48), except that v is replaced by M and S by σ. Relativistic magnetohydro-

dynamics was explored by Achterberg [90] who applied Hamilton’s principle to a

relativistic Lagrangian to yield the same basic laws as those in (3.46), albeit in

relativistic form.

Several authors, most notably Morrison, have continued to apply Hamilto-

nian mechanics and Dirac’s constraint theory to hydrodynamics, magnetohydro-

dynamics and plasma physics in particular, with great success [47, 49, 93, 94].


Several influential papers dealing with noncanonical Hamiltonian methods were

written by Littlejohn as well as Littlejohn & Cary [46, 59, 60]. Littlejohn first

considered a noncanonical Hamiltonian formulation of guiding centre motion [46],

which describes the motion of a charged particle in a magnetic field around a

central ‘guiding’ point which is itself moving. Such motion results in the par-

ticle following a helical orbit through space. This dynamic behaviour is most

certainly applicable to plasma physics, especially in MCF where the plasma is

confined in extremely strong magnetic fields, although it is not further considered

by this author beyond the following review (this thesis focuses on laser-plasma

interaction).

Littlejohn proposed that his Hamiltonian formulation was unique in preserving

Liouville’s theorem and allowing for time averaging of the guiding centre system

78


(the relatively fast gyrating motion of the particle around the guiding centre is

most practically averaged out over one period). He considered a set of coordinates:

the particle position x; time t; kinetic energy k; u the component of velocity

parallel to the field; w the perpendicular velocity; θ the phase of the particle

gyration frequency about the guiding centre. All these coordinates are in the

reference frame moving with the guiding centre.

Note the introduction of time t as an independent variable, which is often

done when the canonical Hamiltonian explicitly depends on time. This‘extended

phase space’ has coordinates and conjugate momenta that now include time and

its conjugate variable, energy (k). The phase space trajectories are then all

parameterised by a new variable τ .

The Hamiltonian (derived by Littlejohn without need of the Legendre trans-

formation from a Lagrangian) is

H(x, t, k, u, w, θ) =1

2(u2 + w2)− k.

Littlejohn introduced the noncanonical Poisson bracket

[F,G] =∂F

∂qi{qi, qj}

∂G

∂qj

where qi represents all eight variables mentioned above and the bracket in the

centre is the usual Poisson bracket. The equations of motion then follow from the

noncanonical Poisson bracket of the generalised coordinates with the Hamiltonian

in the usual way where

∂qi∂τ

= [qi, H].

However, Littlejohn was not concerned with reproducing these equations as

much as applying Darboux’s theorem [54] (see §2.5) to find higher-order terms

in guiding centre motion, although the general concept of applying Hamiltonian

mechanics to guiding centre theory is of the greatest interest for the purposes of

this thesis.

79

3.8 The Korteweg-de Vries Equation

Littlejohn extended this idea to perturbation theory in noncanonical Hamil-

tonian coordinates [59], which was further expanded upon by Cary & Littlejohn

[60] in the context of magnetic field line flow.

Guiding centre theories were also investigated by Pfirsch & Morrison [95, 96]

who sought to find the explicit form of the energy-momentum tensor in this

context. However, in contrast to the material presented by this author in Chapter

4, Pfirsch & Morrison were not able to produce a manifestly gauge invariant

and symmetric tensor from their Lagrangian in [95]. It took some additional

arguments regarding the relation of the energy-momentum tensor to the angular

momentum tensor to prove symmetry. Gauge invariance was proven by splitting

the energy-momentum tensor into a gauge and non-gauge invariant part, and

then showing that the non-gauge invariant part was in fact zero. A linearised

Maxwell-Vlasov and kinetic guiding centre theories was addressed in [96] using a

Hamilton-Jacobi formalism and Dirac’s constraint theory (see §2.5). Brizard &

Tronko also looked at gyrokinetic conservation laws using Hamiltonian mechanics

[97].

An excellent review of these canonical formulations of guiding centre theories

can be found in [98].


This section presents an interesting aside in the area of Hamiltonian fluid dy-

namics, specifically related to the Korteweg-de Vries (KdV) equation. The KdV

equation is a nonlinear wave equation;

∂u

∂t− u∂u

∂x− ∂3u

∂x3= 0.

Other versions of the KdV equation may include certain constant coefficients,

although these are always subject to renormalisation of the function u(x, t). This

equation has solutions that represent solitons or ‘solitary waves’. Solitons are a

80


single wave packet propagating according to the above equation, with the inter-

esting property that they may pass through each other without interacting. They

are encountered in the study of plasma in the form of ion-acoustic waves.

A Hamiltonian formulation of the KdV equation was found by Gardner [99]

in the fourth of a series of papers [100, 101, 102, 103, 104] dedicated to the KdV

equation, its solutions, and its countably infinite conserved quantities.

Gardner found that given a Hamiltonian

H =

∫D

1

6u3 − 1

2

(∂u

∂x

)2

dx

and noncanonical Poisson bracket

{F,G} = −∫D

δF

δu

∂

∂x

δG

δudx,

the KdV equation emerges from Hamilton’s equation of motion for u;

∂u

∂t= {u,H} = −u∂u

∂x− ∂3u

∂x3.

The fact that the KdV equation has countably infinite integrals of motion is

due to the fact that its Hamiltonian formulation is not unique [105].

81

4

The Energy-Momentum Tensor

In Higher-Derivative Theories

4.1 The Gauge Invariant Electromagnetic Energy-

Momentum Tensor

This definition of the energy-momentum tensor creates difficulties in electro-

magnetism where the generalised coordinates are the components of the four-

potential. In this case, recalling Eqn (2.3), the tensor is defined as

T νµ =

∂L

∂Aα,νAα,µ − δνµL,

which is not a gauge invariant expression and therefore lacks the necessary phys-

ical interpretation of the components of the tensor. While this subject is treated

in every standard physics textbook , e.g, [42] [106], the gauge invariance of the

energy-momentum tensor of the free electromagnetic field is always treated after

calculating the tensor, in an ad hoc manner, by choosing to add a suitable diver-

genceless quantity. However, it was pointed out by Munoz [107], and much later

by Correa-Restrepo & Pfirsch [108], that this is not necessary and a careful treat-

ment of the problem can yield a manifestly gauge invariant expression, provided

82

4.1 The Gauge Invariant Electromagnetic Energy-Momentum Tensor

that the Lagrangian itself is gauge invariant.

The following derivation of the free electromagnetic field energy-momentum

tensor follows Munoz, but it will be extended by this author in §4.2 to include

Lagrangians with derivatives of higher-order. Consider the action

S =

∫L(Aµ, Aµ,ν)d

4x. (4.1)

The Principle of Least Action gives the usual Euler-Lagrange equations

∂L

∂Aµ=

∂

∂xα

(∂L

∂Aµ,α

).

Consider now an infinitesimal transformation of the coordinates such that

x′τ = xτ + ωτσxσ + aτ = xτ + ετ (x)

where ωτσ represents a Lorentz transformation (and so is necessarily an anti-

symmetric tensor) and aτ is an infinitesimal local translation of the coordinates.

With this transformation,

A′µ =∂xα

∂x′µAα = (δαµ −

∂εα

∂x′µ)Aα = (δαµ −

∂xβ

∂x′µ

∂εα

∂xβ)Aα. (4.2)

This expression can be evaluated to any order in

∂xβ

∂x′µ,

given the appearance of this expression again on the right hand side. Taken to

first order, we have

A′µ = (δαµ − εα,µ)Aα. (4.3)

Similarly, for first-order derivatives of the potential,

A′µ,ν = (∂ν − εβ,ν∂β)(δαµ − εα,µ)Aα (4.4)

83


A change of coordinates in Eqn (4.1) gives

S ′ =

∫L(A′µ(x′),

∂

∂x′νA′µ(x′))d4x′

=

∫L(Aµ − εα,µAα, (∂ν − εβ,ν∂β)(δαµ − εα,µ)Aα

)d4x

If the variation of S is now calculated using the fact that the variation in the

coordinates Aµ is given by the infinitesimal transformation part of Eqn (4.3) to

first order in ε, then

δS = S ′ − S = −∫ (

∂L

∂Aµεα,µAα +

∂L

∂Aµ,ν(εα,µAα,ν + εα,νAµ,α)

)d4x. (4.5)

If the Lagrangian itself is gauge invariant, as in the case of the free electro-

magnetic field (but not where there is an interaction term for a particle in a field),

then the derivatives of the Lagrangian with respect to the field must be gauge

invariant and appear only in the combination

Aµ,ν − Aν,µ.

In this case, it is true that

∂L

∂Aµ,ν= − ∂L

∂Aν,µ.

Eqn (4.5) then becomes

δS = −∫ (

∂L

∂Aµεα,µAα −

∂L

∂Aν,µεα,µAα,ν +

∂L

∂Aµ,νεα,νAµ,α

)d4x.

84


Now a simple relabelling of the dummy indices µ, ν in the second term gives

δS = −∫ (

∂L

∂AνAα +

∂L

∂Aµ,ν(Aµ,α − Aα,µ)

)εα,νd

4x

= −∫∂ν

((∂L

∂AνAα +

∂L


)εα)d4x

+

∫∂ν

(∂L

∂AνAα +

∂L


)εαd4x

Call the expression inside the brackets in the second term I να and evaluate

its divergence using: the Euler-Lagrange equations; independence of L from Aµ;

antisymmetry of the derivatives with respect to Aµ,ν . This yields

I να,ν =

∂

∂xν

(∂L

∂Aν

)Aα +

∂L

∂AνAα,ν +

∂

∂xν

(∂L

∂Aµ,ν

)(Aµ,α − Aα,µ)

+∂L

∂Aµ,ν(Aµ,αν − Aα,µν)

=∂

∂xν

(∂L

∂Aµ,νAµ,α

)− ∂L

∂Aµ,νAα,µν

=∂

∂xν

(∂L

∂Aµ,νAµ,α

)

If the derivation that led to Eqn (2.3) were repeated with respect to a partial

derivative of L, then the result would be

∂L

∂xα=

∂

∂xν

(∂L

∂Aµ,νAµ,α

)= I ν

α,ν .

The variation of the action is then

δS = −∫ (

∂ν

((∂L

∂AνAα +

∂L


)εα − ενL

))d4x

since εν,ν = 0 due to the antisymmetry of the Lorentz transformation ωντ . The ap-

plication of Noether’s theorem is now a simple matter, and the conserved current

85

4.2 Lagrangians with Higher-Order Derivatives

is

Jν =

(∂L

∂AνAα +

∂L


)εα − ενL

where the energy-momentum tensor belongs to the translational part of ε, and the

angular momentum tensor to the rotational part (the Lorentz transformation).

A manifestly gauge invariant expression for the energy-momentum tensor is then

simply expressed as

T να = 2∂L

∂FνµFαµ − δναL (4.6)

since

∂L

∂Aν= 0

for a gauge invariant Lagrangian and derivatives with respect to the four-potential

can be replaced with derivatives with respect to the electromagnetic tensor;

∂L

∂Aα,β=

∂Fτω∂Aα,β

∂L

∂Fτω(4.7)

= (δβτ δαω − δατ δβω)

∂L

∂Fτω

= 2∂L

∂Fβα.


The derivation of the canonical energy-momentum tensor (2.3) can be generalised

for higher derivatives, as can the Euler-Lagrange equations themselves. More

generally, the Euler-Lagrange equations can be found as a functional derivative

of the action with respect to the generalised coordinates qi. Such a functional

derivative includes all higher derivatives of the generalised coordinates that may

appear in the Lagrangian (and therefore the action), and is given by

δS

δqi=

∫ (∂L

∂qi− d

dxα

(∂L

∂qi,α

)+

d2

dxαdxβ

(∂L

∂qi,αβ

)− . . .

)d4x.

If the variation of S is zero for any arbitrary δqi, then the above expression

86


implies the higher-derivative Euler-Lagrange equations

=⇒ ∂L

∂qi=

d

dxα

(∂L

∂qi,α

)− d2

dxαdxβ

(∂L

∂qi,αβ

)+ . . . (4.8)

Therefore, if the Lagrangian includes second-order derivatives of the gener-

alised coordinates, the derivation of the energy-momentum tensor follows the

same procedure as in §4.1, but this time substituting (4.8);

dL

dxµ=∂L

∂qiqi,µ +

∂L

∂qi,νqi,νµ +

∂L

∂qi,βνqi,βνµ +

∂L

∂xµ

=d

dxν

(∂L

∂qi,ν

)qi,µ −

d2

dxνdxβ

(∂L

∂qi,νβ

)qi,µ +

∂L

∂qi,νqi,νµ +

∂L

∂qi,βνqi,βνµ +

∂L

∂xµ

=d

dxν

(∂L

∂qi,νqi,µ

)− d

dxν

(d

dxβ

(∂L

∂qi,νβ

)qi,µ

)+

d

dxβ

(∂L

∂qi,νβ

)qi,µν

+d

dxν

(∂L

∂qi,βνqi,βµ

)− d

dxν

(∂L

∂qi,βν

)qi,βµ +

∂L

∂xµ

=d

dxν

(∂L

∂qi,νqi,µ −

d

dxβ

(∂L

∂qi,νβ

)qi,µ +

∂L

∂qi,βνqi,βµ

)+∂L

∂xµ

=⇒− ∂L

∂xµ=

d

dxν

(∂L

∂qi,νqi,µ −

d

dxβ

(∂L

∂qi,βν

)qi,µ +

∂L

∂qi,βνqi,βµ − δνµL

).

The energy-momentum tensor is then

T νµ =∂L

∂qi,νqi,µ −

d

dxβ

(∂L

∂qi,βν

)qi,µ +

∂L

∂qi,βνqi,βµ − δνµL. (4.9)

However, if the generalised coordinates qi represent the components of a

vector-field, as in the case of the electromagnetic potential, then the above ex-

pression is not manifestly gauge invariant. To remedy this, the same procedure

demonstrated by Munoz which was reviewed in §4.1 will now be applied, for the

first time, to a gauge invariant Lagrangian dependent on second-order derivatives

of the electromagnetic potential. Consider the action integral

S =

∫L(Aµ, Aµ,ν , Aµ,ντ )d

4x

87


subject to an infinitesimal transformation of coordinates such that

x′τ = xτ + ωτσxσ + aτ = xτ + ετ (x),

then

∂2A′(x′)µ∂x′τ∂x′ν

=∂xα

∂x′µ

∂xβ

∂x′ν

∂xγ

∂x′τAα,βγ

= (δαµ −∂εα

∂x′µ)(δβν −

∂εβ

∂x′ν)(δγτ −

∂εγ

∂x′τ)Aα,βγ.

According to Eqns (4.2) and (4.3), this gives, to first order in ε,

∂2A′(x′)µ∂x′τ∂x′ν

= (δαµ − εα,µ)(δβν − εβ,ν)(δγτ − εγ,τ )Aα,βγ

= (δαµδβν δ

γτ − δαµδβν εγ,τ − δβν δγτ εα,µ − δαµδγτ εβ,ν)Aα,βγ

= Aµ,ντ − εγ,τAµ,νγ − εα,µAα,ντ − εβ,νAµ,βτ .

The variation of the action given by S ′ − S is then

δS = −∫ (

∂L

∂Aµεα,µAα +

∂L

∂Aµ,ν(εα,µAα,ν + εα,νAµ,α)

)d4x

−∫ (

∂L

∂Aµ,ντ(εγ,τAµ,νγ + εα,µAα,ντ + εβ,νAµ,βτ )

)d4x.

Relabelling some dummy indices gives

δS = −∫ (

∂L

∂AµAα +

∂L

∂Aµ,νAα,ν +

∂L

∂Aν,µAν,α

)εα,µd

4x (4.10)

−∫ (

∂L

∂Aτ,νµAτ,να +

∂L

∂Aµ,ντAα,ντ +

∂L

∂Aν,µτAν,ατ

)εα,µd

4x.

88


Since the Lagrangian is gauge invariant, derivatives of L with respect to

derivatives of the four-potential can be replaced by derivatives with respect to

the electromagnetic tensor as follows:

∂L

∂Aα,βγ=

∂F τω,ω

∂Aα,βγ

∂L

∂F τε,ε

(4.11)

=∂(Aω,τω − Aτ,ωω)

∂Aα,βγ

∂L

∂F τε,ε

=∂L

∂F ,εβε

gαγ − ∂L

∂F ,εαεgγβ.

While it is clear from the above expression that

∂L

∂Aα,βγ= − ∂L

∂Aβ,αγ,

extreme care must still be taken with the order of the four-potential derivatives

appearing in such expressions, since they also satisfy a kind of Jacobi identity

whereby

∂L

∂Aα,βγ+

∂L

∂Aγ,αβ+

∂L

∂Aβ,γα= 0.

Consider the terms inside the brackets in (4.10) and call them I µα collectively,

so that

δS =

∫ (I µα,µε

α − ∂

∂xµ(I µα ε

α)

)d4x. (4.12)

This can be cast into a form allowing the application of Noether’s theorem as

follows. First, consider the partial derivative of the Lagrangian,

∂L

∂xα=

∂L

∂AνAν,α +

∂L

∂Aν,µAν,µα +

∂L

∂Aν,µτAν,µτα.

Substituting the Euler-Lagrange equations in the first term, then simplifying,

gives

∂L

∂xα=

∂

∂xµ

(∂L

∂Aν,µAν,α −

∂

∂xτ

(∂L

∂Aν,µτ

)Aν,α +

∂L

∂Aν,τµAν,τα

).

89


Comparing the above expression with I µα,µ, we have

∂L

∂xα= I µ

α,µ −∂

∂xµ

(∂L

∂AµAα +

∂L

∂Aµ,νAα,ν +

∂L


∂L

∂Aν,µτAν,ατ

)− ∂

∂xµ

(∂

∂xτ

(∂L

∂Aν,µτ

)Aν,α

).

Using the Euler-Lagrange equations again, the antisymmetry of the derivatives

of L with respect to the four-potential derivatives, and the Jacobi identity for

derivatives with respect to the second order derivatives of the four-potential, the

above expression simplifies to

I µα,µ =

∂L

∂xα+

∂2

∂xµ∂xτ

(∂L

∂Aµ,ντFνα

).

The second order derivative of ε vanishes and εα,α = 0, so therefore

I µα,µε

α =∂Lεα

∂xα+

∂

∂xµ

(∂

∂xτ

(∂L

∂Aµ,ντFνα

)εα − ∂L

∂Aτ,νµFναε

α,τ

)

The third term in the expression above can be expressed with a factor of ε, and

not its derivative, by use of Eqn (4.4), which gives:

I µα,µε

α =∂Lεα

∂xα+

∂

∂xµ

(∂

∂xτ

(∂L

∂Aµ,ντFνα

)εα +

∂L

∂Aτ,νµ(A′ν,τ − Aν,τ )

)(4.13)

=∂Lεα

∂xα+

∂

∂xµ

(∂

∂xτ

(∂L

∂Aµ,ντFνα

)εα +

1

2

∂L

∂Aτ,νµFντ,αε

α

).

Now let

I µα =

∂L

∂AµAα + 2

∂L

∂FνµFνα +K µ

α (4.14)

90


where

K µα =

∂L

∂Aτ,νµAτ,να +

∂L


∂L

∂Aν,µτAν,ατ

=1

2

∂L

∂Aτ,νµFντ,α +

∂L

∂Aµ,ντFνα,τ

Substituting (4.13) and (4.14) into (4.12), and then applying (4.11), finally gives

the variation of the action as

δS = −∫

∂

∂xµ

((∂L

∂AµAα + 2

∂L

∂FνµFνα

)εα)d4x

−∫

∂

∂xµ

((− ∂

∂xµ

(∂L

∂F ,βνβ

)Fνα +

∂

∂xν

(∂L

∂F ,βµβ

)Fνα − δµαL

)εα

)d4x

The conserved current is therefore

Jµ =

(∂L

∂AµAα + 2

∂L

∂FνµFνα −

∂

∂xµ

(∂L

∂F ,σνσ

)Fνα +

∂

∂xν

(∂L

∂F ,σµσ

)Fνα − δµαL

)εα

with the translational part of ε giving the energy-momentum tensor. So the final

gauge invariant energy-momentum tensor for a gauge invariant electromagnetic

Lagrangian dependent on second-order derivatives is

T νµ = 2∂L

∂FναFµα −

∂

∂xν

(∂L

∂F ,σασ

)Fαµ +

∂

∂xα

(∂L

∂F ,σνσ

)Fαµ − δνµL. (4.15)

However, if the Lagrangian itself is not gauge invariant (as in the case where

there is an interaction term for a charged particle, JαAα), then (4.15) can still

be used to simplify calculations by splitting it into a gauge invariant part and

non-gauge invariant part. The non-gauge invariant part must then be dealt with

using the canonical expression (4.9) such that

T νµ =∂L

∂Aα,νAα,µ −

d

dxβ

(∂L

∂Aα,βνAα,µ

)+ 2

∂L

∂Aα,βνAα,βµ − δνµL. (4.16)

This thesis has therefore demonstrated, for the first time, that the rigorous

91


approach of Munoz is applicable to a second order theory and yields an expression

for the energy-momentum tensor that is manifestly gauge invariant.

It is well worth mentioning that Lagrangians with higher derivatives were orig-

inally studied by Ostrogradski [109] who considered the stability of the Hamil-

tonian corresponding to such a Lagrangian. While Ostrogradski’s work has seen

some application in the world of physics [110, 111, 112], it will not be further

explored in this thesis beyond the following precis of Ostrogradski’s work.

Ostrogradski showed that if the highest time derivative of the Lagrangian is

non-degenerate, the Hamiltonian will have at least one linear instability [112].

The term non-degenerate means that the equations that give the canonical mo-

mentum Pi,

Pi =δL

δqi,

can be inverted to give qi in terms of the canonical coordinates only. Otherwise,

the Legendre transform of the Lagrangian (the Hamiltonian),

H = Piqi − L(qi, qi),

could not be expressed purely in terms of canonical coordinates Qi, Pi. The linear

instability refers to the appearance of a term linear in the canonical momentum

in the Hamiltonian, rather than quadratic as usual, therefore leaving the Hamil-

tonian (energy) unbounded from below.

While Ostogradski’s theorem is still be a focus of study for any physicist ex-

ploring Hamiltonian descriptions of nature, the following section will deal purely

with the Lagrangian side of the higher-derivatives problem. In particular, the

Podolsky Lagrangian will be substituted into (4.15) to give a gauge invariant

expression for the Podolsky energy-momentum tensor.

92

4.3 The Gauge Invariant Energy-Momentum Tensor for the PodolskyLagrangian

4.3 The Gauge Invariant Energy-Momentum Ten-

sor for the Podolsky Lagrangian

Possibly the most well-known example of a Lagrangian theory of fields with higher

derivatives is that of Podolsky electrodynamics. Podolsky considered Ostragrad-

ski’s approach applied to electromagnetism and found that there was only one

possible generalisation of the electromagnetic Lagrangian that would yield linear

field equations below sixth-order [7];

L = − 1

4µ0

FαβFαβ +

1

2µ0

a2Fαβ,βF

,γαγ (4.17)

where a is some new constant of nature with dimensions of length. This constant

will be addressed in more detail toward the end of this section, but for now it is

sufficient to say that this constant must be very small, otherwise the deviations

from Maxwell’s theory predicted by Podolsky would have been detected already.

Cuzinatto et al. [113] showed that, to be consistent with quantum theory and

experiments, a ≤ 5.6 × 10−15m - the order of the Compton wavelength of the

electron [114].

That (4.17) is the only possible modification of the electromagnetic Lagrangian

(giving linear differential equations below sixth-order) can be seen by considering

the underlying nature of the electromagnetic tensor in terms of the electric and

magnetic fields. Podolsky found that the extra term in (4.17) is the only other

Lorentz invariant that can be constructed with the electromagnetic tensor that

does not make a vanishing contribution to the field equations (while E ·B is also a

Lorentz invariant, it can be expressed as the four-divergence of the four-current,

and would therefore not contribute anything to the equations of motion). The

uniqueness of (4.17) has also been re-examined and confirmed recently [115].

The Podolsky Lagrangian keeps Maxwell’s electrodynamics largely intact in

that the Euler-Lagrange equations are simply generalisations of Maxwell’s equa-

tions that take into account higher-derivatives. The Lorentz force is also un-

93


changed by this Lagrangian, since it is derived from the kinetic energy and in-

teraction term in the Lagrangian, which can be added as usual to (4.17). What

then is the point of the Podolsky Lagrangian? The critical difference between

Podolsky’s Lagrangian and the standard electromagnetic Lagrangian is that it

gives a finite energy for a charged point particle without any further tinkering

or renormalisation. It also better describes waves of very high frequency where

derivatives of the field can no longer be considered negligible, and can be effec-

tively quantized [116].

The Euler-Lagrange equations for the Lagrangian (4.17) are

δL

δAµ= 0 =⇒ ∂L

∂Aµ=

∂

∂xτ

(∂L

∂Aµ,τ

)− ∂2

∂xωxτ

(∂L

∂Aµ,τω

)

which gives (1 + a2

∂2

∂xγxγ

)F ,τµτ = 0 (4.18)

Converting F ,τµτ to vector notation, Eqn (4.18) is more easily recognisable as

a generalised version of Maxwell equations in free space:

(1 + a2

∂2

∂xγ∂xγ

)(∇×B− 1

c2∂E

∂t

)= 0; (4.19)

(1 + a2

∂2

∂xγ∂xγ

)∇ · E = 0. (4.20)

Including the current and charge densities on the right hand side of Eqn (4.19)

and (4.20), respectively, only requires the addition of an interaction term JαAα

to the Lagrangian, where Jα is the four-current. In electrostatics,

E = −∇φ,

and since the electric field does not change, the second order time derivative in

Eqn (4.20) can be ignored. The generalised Poisson’s equation for a point charge

94


qδ(r), where δ is the Dirac-delta functional, is then

(1 + a2∇2)∇2φ = − 1

ε0qδ(r).

This equation has only one solution of the form

φ =q(1− e−r/a)

ε0r

where r is now the scalar radial distance from the origin. Note that by application

of l’Hopital’s Rule,

limr→0

q(1− e−r/a)ε0r

= limr→0

q

ε0ae−r/a =

q

ε0a,

indicating that the self-energy of a charged point particle has a finite value at

q2/ε0a, unlike standard electrodynamics where the energy diverges as r → 0.

The Lagrangian (4.17) will now be subsituted into Eqn (4.15 which gives the

energy-momentum tensor for Podolsky electrodynamics:

T νµ = Tνµ +a2

µ0

(−Fασ ν

,σ F µα + F νσ α

,σ F µα − gνµ

1

2Fαβ

,βF,γ

αγ

)(4.21)

where Tνµ is the usual Maxwell energy-momentum tensor for the unmodified

Lagrangian [106],

Tνµ =1

µ0

(−F ναF µα + gνµ

1

4FαβF

αβ).

The gauge invariant energy-momentum tensor (4.15) was derived in general for

any gauge invariant Lagrangian and when applied to the Podolsky Lagrangian

it yielded (4.21) - a tensor that is already gauge invariant, in contrast to that

derived by Podolsky himself [7] [117] [118].

To satisfy the reader that (4.21) is indeed correct, its equivalence to the origi-

nal Podolsky tensor will be demonstrated. Note that the second and third terms

can be combined via the Jacobi identity for the electromagnetic tensor such that

95


−Fασ ν,σ F µ

α + F νσ α,σ F µ

α = F να,σσF

µα . (4.22)

Now consider an odd expression that appears to come from nowhere, but will

provide a useful (but not obvious) identity when the Podolsky Lagrangian (4.17)

is inserted;

∂L

∂Fντ,αFατ,µ =

1

2

∂L

∂Fντ,αFατ,µ −

1

2

∂L

∂Fτν,αFατ,µ (4.23)

F νσ,σF

ττ,µ =

1

2F νσ

,σFττ,µ +

1

2F τσ

,σFντ,µ

=⇒ ∂

∂xµ(F ,σ

ασ Fαν)− F ,σµ

ασ Fαν = F ,σασ F

αν,µ = 0 (since Fαα = 0).

The divergence of the first term of the last line above is

∂2

∂xν∂xµ(F ,σ

ασ Fαν) =

∂2

∂xβ∂xµ(F ,σ

ασ Fαβ) =

∂2

∂xν∂xβ(gµνF ,σ

ασ Fαβ)

=⇒ ∂

∂xµ(F ,σ

ασ Fαν) =

∂

∂xβ(gµνF ,σ

ασ Fαβ).

Also, note that

gµνF σαβ,σ F

αβ = −gµνF σσα,β F

αβ − gµνF σβσ,α F

αβ

= 2∂

∂xβ(gµνF ,σ

ασ Fαβ)− 2gµνF ,σ

ασ Fαβ,β

which means that

∂

∂xµ(F ,σ

ασ Fαν) =

∂

∂xβ(gµνF ,σ

ασ Fαβ) = gµν(

1

2F σαβ,σ F

αβ + F ,σασ F

αβ,β). (4.24)

Substituting (4.24) for the first term of the last line in (4.23), and using the

96


Jacobi identity for the electromagnetic tensor on the second term, gives

F ,σασ F

αν,µ = gµν(1

2F σαβ,σ F

αβ + F ,σασ F

αβ,β) + F µα,σ

σFνα + F σµ,α

σFνα

= gµν(1

2F σαβ,σ F

αβ + F ,σασ F

αβ,β)− F µα,σ

σFνα +

∂

∂xα(F µσ

,σFνα)

− F µσ,σF

,ανα (4.25)

The fourth term in the second line above is divergenceless, so substituting

(4.22) and (4.25) (which is equal to zero and does not affect the equations of

motion) back into (4.21) gives the symmetric energy-momentum tensor as it was

originally derived by Podolsky:

T νµ = Tνµ +a2

µ0

(gµν1

2(F σ

αβ,σ Fαβ + F ,σ

ασ Fαβ,β)− F να,σ

σFµα − F µα,σ

σFνα

− F νσ,σF

µα,α). (4.26)

There are much simpler ways to make Eqn (4.21) symmetric, but the goal here was

to prove an exact match to the symmetric version originally derived by Podolsky

who used a different method to construct the tensor.

Since the focus of this thesis is on plasma physics, an investigation of the

parameter a and the dispersion relation for waves in a plasma modeled on the

Podolsky Lagrangian will be considered. Would the Debye length then be an

appropriate value to assume for the constant a, or is there another way to derive

a for laser-plasma interaction without further assumption?

Consider that if a current is present then Eqn (4.19) is

(1 + a2

∂2

∂xγxγ

)(∇×B− 1

c2∂E

∂t

)= µ0J.

Even though the Podolsky Lagrangian contains higher derivatives of the field,

it assumed no modification to the particle and particle/field interaction terms in

the Lagrangian. In this case, the Lorentz force is unchanged and the generalised

97


Ohm’s law for a plasma, Eqn (1.10) derived in §1.2.5 is still applicable where

J = iε0ω2p

ωE.

A new plasma dispersion relation then results from

(1 + a2

∂2

∂xγxγ

)(∇×B− 1

c2∂E

∂t

)= i

ω2p

c2ωE

where E and B are assumed proportional to ei(k·r−ωt) as in §1.2.5. The dispersion

relation is (k2 − ω2

c2

)(1 + a2

(k2 − ω2

c2

))= −

ω2p

c2, (4.27)

which can be rearranged to give a in terms of the variables ωp, k and ω

a2 =−ω2

p − c2k2 + ω2

c2(k2 − ω2/c2)2.

Alternatively, it could also be expressed in the same form as Eqn (1.4) with

(squared) dimensions of speed over frequency;

a2 =c2 + ω2

p/k2 − ω2/k2

−c2/k2(k2 − ω2/c2)2. (4.28)

Eqn (4.28) has been presented for the first time in this thesis, and no-one

else has attempted deriving the Podolsky constant in this way in the context of

laser-plasma interaction.

An analysis of the above expression can yield values for the frequencies that

could probe Podolsky electrodynamics. It is clear that the numerator can be

at most of order 9 × 1016m2s−2 since ω > ωp or the wave would be cut off. It

was already mentioned at the beginning of this section that a ≤ 5.6 × 10−15m.

Therefore, to be consistent with the femtometre scale of a, the denominator must

be ≥ 2.9 × 1045s−2, which corresponds to an extremely high energy wave in the

gamma ray range with a frequency of ≈ 5.4× 1022s−1 or higher. Therefore, high

98


intensity laser-plasma interaction may provide a testbed for the predictions of

Podolsky electrodynamics.

Note that in Maxwell’s electrodynamics, ω2 = c2k2, which is meaningless

in Podolsky electrodynamics since a would be undefined. That the dispersion

relation for electromagnetic waves is altered in Podolsky electrodynamics, such

that ω2 6= c2k2, implies that photons have mass. Of course, this mass must be

extremely small to fit experimental data which has set an upper mass on the

photon at between 7× 10−17eV/c2 [119] and 3× 10−27eV/c2 [120], depending on

the model used and assumptions made about the photon itself.

Contrary to popular belief, a massive photon does not preclude gauge invari-

ance in electrodynamics [121]. Podolsky Electrodynamics is a gauge and Lorentz

invariant theory that incorportates photon mass. However, most authors overlook

this since the Proca Lagrangian, given by

L = − 1

4µ0

FαβFαβ + µAαA

α,

is more well-known than the Podolsky Lagrangian and is usually the model used

to incorporate photon mass µ into electrodynamics (at the expense of gauge

invariance).

A link has been previously made between Podolsky electrodynamics and

plasma physics by Santos [122]. He noted that the modified Podolsky disper-

sion relation (for free space),

(k2 − ω2

c2

)(1 + a2

(k2 − ω2

c2

))= 0, (4.29)

indicates that the electromagnetic waves are travelling through a tenuous plasma

even though the free space solution is being considered. To see this, recall Eqn

(1.4) for the Debye length derived in §1.2.2,

λD =vavgωp

.

99


Given that the dispersion relation in (4.29) was derived for vacuum, the av-

erage velocity in the equation above can be identified with c while the constant

a (with dimensions of length) can be identified with the Debye length, giving an

expression for the frequency of a plasma in ‘free space’,

a =c

ωp.

In this way, Santos assumed a form for the constant a, as opposed to the

method presented in this thesis that led to Eqn (4.27) by deriving it from first

principles. In this case, Eqn (4.29) gives two modes for wave propagation in free

space; the standard free space dispersion relation

ω = ck

and the plasma dispersion relation

ω2p = ω2 − c2k2.

Santos believes this plasma relation to be due to vacuum polarization which

creates a particle density given by

n =ε0mω

2p

q2=

m

µ0q2a2≈ 760× 1036m−3

where a = λC/2. Given that the reduced Compton wavelength λC of an electron

is approximately 3.86× 10−13m [114], a sphere of radius λC has a total volume of

roughly 2.409× 10−37m3 which would contain on average around 183 virtual par-

ticles according to Podolsky electrodynamics. Santos believes that the creation

and annihilation of these particles contributes to the Zitterbewegung, or electron

‘jitter’ motion predicted by quantum mechanics at distances of the order of the

Compton wavelength [123].

Whether either Santos or this author (or both) are correct in their analysis of

100


Podolsky electrodynamics is a question that can only be settled by experiment.

However, the point to emphasise here is that the length and energy scales dis-

cussed in this section are in the realm of high intensity laser-plasma interactions

encountered in ICF experiments. This thesis posits that the seemingly small

modification of the electromagnetic Lagrangian introduced by Podolsky will in

fact yield more accurate predictions in laser-plasma interaction.

101

5

The Nonlinear Ponderomotive

Force in Laser-Plasma Interaction

5.1 The Ponderomotive Force

A ponderomotive force is usually defined as a force on a charged particle due to

an oscillating electric field, expressed as

f = − q2

4mω2∇E2

where ω is the oscillation frequency of the field and m the mass of the particle.

This force was first observed in experiment in 1957 [124].

The derivation of the ponderomotive force is reasonably straightforward and

uses the Lorentz force with the assumption of a varying electric field amplitude.

Consider first the Lorentz force for an electromagnetic field oscillating with fre-

quency ω such that

mdv

dt= q(E + v ×B) cosωt.

Neglecting the magnetic field for the moment gives a first approximation of

the particle velocity as

v(1) =qE sinωt

mω(5.1)

102


and position as

r(1) = −qE cosωt

mω2. (5.2)

These approximations will be used as first order variations in such a way that

a small variation in the electric field amplitude is given by

δE = δxi∂E

∂xi= (r(1) · ∇)E,

which also has the effect of altering the velocity where

δv = v(1).

With this variation in the electric field amplitude, the Lorentz force is given

by

f = q((r(1) · ∇)E + v(1) ×B) cosωt (5.3)

= −q2 cos2 ωt

mω2(E · ∇)E +

q2 sinωt cosωt

mωE×B

Although the field is oscillating, the cumulative effect of the force on a charged

particle over time is of real interest. Consider the average of cos2 ωt over time;

〈cos2 ωt〉 = limT→∞

1

T

∫ T

0

(1

2cos 2ωt+

1

2)dt

= limT→∞

1

T

[1

4ωsin 2ωt+

t

2

]T0

=1

2.

The same procedure gives

〈sinωt cosωt〉 = 〈12

sin 2ωt〉 = 0,

103


and so the time averaging of Eqn (5.3) gives

f = − q2

2mω2(E · ∇)E.

Using the vector calculus identity

E · ∇E =1

2∇E2 − E× (∇× E)

and the Maxwell-Faraday equation

∇× E = −∂B

∂t

gives

f = − q2

2mω2

(1

2∇E2 + E× ∂B

∂t

).

However, considering again the time average of this expression means that

B does not change in time, so the second term above is zero, finally giving the

Ponderomotive force as

f = − q2

4mω2∇E2. (5.4)

Interestingly, the direction of this force does not depend on the sign of the

charged particle itself. This means that in a plasma, both electrons and ions

would be accelerated in the same direction by the ponderomotive force, although

the much greater mass of the ions limits their acceleration relative to the plasma

electrons. Note that the ponderomotive force causes charged particles to drift

toward areas of low field intensity, pushing them outward from a laser beam.

This is in contrast to similar force derivations applied to dielectric molecules in a

laser beam which are accelerated toward regions of higher intensity according to

f =1

2α∇E2,

104

5.2 The Nontransient Nonlinear Ponderomotive Force

where the polarisation P of the molecule satisfies the linear relation

P = αE

and α has the same dimensions as ε0. In this case, the molecules are trapped in

the laser beam which acts as an ‘optical tweezer’ [125].

The significance of the ponderomotive force in relation to high intensity laser

beams was explored by Quesnel and Mora [126]. They showed that at high

intensities, the force must be generalised to a relativistic ponderomotive force. A

generalisation of the transient nonlinear ponderomotive force will be discussed in

§5.3.

5.2 The Nontransient Nonlinear Ponderomotive

Force

The nonlinear ponderomotive force is particularly relevant in the context of iner-

tial confinement fusion. Indeed, the very promising side-on ignition fusion scheme

has a theoretical foundation in the nonlinear force [127, 128]. The prospect of

fusion energy production with less radioactivity than burning coal (even coal

contains trace amounts of uranium) via Hydrogen/Boron-11 side-on ignition is

exciting given recent calculations suggesting that this reaction would be only one

order of magnitude more difficult than traditional Deuterium/Tritium reactions

using spherical laser compression [129, 130].

It was shown by Schluter [131] that a simple two-fluid model of a plasma

could yield an equation of motion for the plasma current in terms of the pon-

deromotive force. An in-depth discussion of this can be found in [18], bu the

essential argument uses the two-fluid model of a plasma. Assuming only one ion

species in the plasma, ions and electrons can be individually treated as charged

fluids. The equations of motion for each plasma particle species is then given by

105


a combination of the Euler equation with the Lorentz force. In the case of ions,

the equation of motion in vector notation is

minidvidt

= −ZniqE− niZqvi ×B−∇(nikBTi)−mneνei(vi − ve) (5.5)

where the subscript i denotes that these quantities refer to the plasma ion fluid.

The parameters νei and Z refer to the electron-ion collision frequency and ion

charge number, respectively. The gasdynamic pressure of the ion species is given

by Pi = nikBTi. A similar equation holds for the plasma electrons considered as

a fluid;

menedvedt

= neqE + neqve ×B−∇(nekBTe) +mneνei(vi − ve). (5.6)

If quasi-neutrality is assumed so that ne ≈ Zni, then the net velocity of the

ion and electron fluids is given by

v ≈ mivi + Zmvemi + Zm

and the average total pressure is

P = nikBTi + nekBTe ≈ ni(1 + Z)kBT

where the ions and electrons are assumed at the same temperature (alternatively,

without this assumption, simply define P ≈ nikB(Ti+ZTe)). The current density

is

J = q(neve − Znivi).

Using the above substitutions, adding Eqns (5.5) and (5.6) and ignoring terms

with me/mi << 1 gives [131] [9]

f = minidv

dt= −∇P + J×B + ε0

ω2p

ω2E · ∇E. (5.7)

106


Thus, the net force derived from a two-fluid description of a plasma yields

a nonlinear term in the electric field. The entire force expression given by Eqn

(5.7) will be referred to as the nonlinear ponderomotive force, although a more

general expression will be introduced shortly.

Adding the fluid equations for ions and electrons yielded the equation of mo-

tion in Eqn (5.7), subtracting Eqn (5.5) from Eqn (5.6), gives

m

q2ne

(dJ

dt+ νJ

)= E + v ×B +

1

qneJ×B +

1

qne

∇P1 + 1/Z

(see Appendix C of [18]) for a detailed derivation). Recalling Eqn (1.3), a gener-

alised Ohm’s Law for the plasma is found when the last three terms are neglected

(that is, the electric field is considered the most dominant contribution in a first

approximation);

dJ

dt+ νJ = ε0ω

2pE. (5.8)

The Lorentz theory of plasma is then described by a system of three equations

- two of Maxwell’s equations in vacuum and the integral of Eqn (5.8) where J

and E are assumed proportional to ei(k·r−ωt):

∇× E = −1

c

∂B

∂t

∇×B = µ0J +1

c2∂E

∂t

J = −ε0ω

2p

iω(1 + iν/ω)E.

In this way, the dynamics of laser-plasma interaction can be neatly described

by Maxwell’s microscopic equations (‘free space’) coupled with a particular form

of Ohm’s law that determines the specific physics for the case of a plasma.

While the previous discussion centred on the application of the two-fluid

model, an expression for the nonlinear ponderomotive force in a nondispersive

fluid dielectric was derived by Landau & Lifshitz [132] in terms of a stress tensor

such that the force density f was given as the divergence of the stress tensor σik

107


in the usual way;

fi =∂σik∂xk

.

Landau & Lifshitz themselves stated that the derivation of the ponderomotive

force for fluid dielectrics is rather complicated. Their derivation involved consid-

ering small variations in fluid position in the presence of a uniform electric field.

The specific form of the tensor derived by Landau & Lifshitz was

σik = −δikε0(P +

(εr − ρ

∂εr∂ρ

)E2

)+ ε0εrEiEk (5.9)

where P is the thermokinetic pressure, ρ the fluid density and εr the relative

permittivity of the fluid. In vacuum, εr = 1 and P = 0 and this tensor becomes

the Maxwell stress tensor from Eqn (2.5) in §2.2 (for zero magnetic field). Af-

ter taking the divergence of this tensor and doing some further manipulations,

Landau & Lifshitz showed that nonlinear ponderomotive force density is

f = −∇P + (ε− 1)∇E2 (5.10)

where ε = ε0εr is the total permittivity. Of course, a more general expression

including magnetic fields would include the time derivative of the Poynting vector

such that

f = ∇ · σ − ε0∂E×B

∂t.

It was Hora [9] who, motivated by Eqn (5.10), demonstrated the equivalence

of a generalised version of Schluter’s Eqn (5.7),

f = J×B + ε0E∇ · E + ε0∇ · (η2 − 1)E⊗ E, (5.11)

with

f = ∇ · (U + ε0(η2 − 1)E⊗ E)− ε0

∂E×B

∂t, (5.12)

where U is the Maxwell stress tensor in Eqn (2.5). A complete derivation of (5.12)

108


from (5.11 can be found in [18]. Note that the appearance of the permittivity in

Eqn (5.10) has been superseded by the refractive index in Eqn (5.12) in accordance

with Eqn (1.18) where η2 = εr when µr = 1. The pressure has also been neglected

as it is the nonlinear terms that are of real interest.

Hora showed that Eqn (5.12) not only contains Schluter’s ponderomotive term

from Eqn (5.7) but is also more general in that it includes spatial derivatives of the

refractive index [9] (required for inhomogeneous plasmas). It is remarkable that

Eqn (5.10), which was derived for nondispersive media, was successfully shown

to be applicable to plasmas with dispersion and absorption by Hora’s application

of the test of momentum conservation for an obliquely incident laser beam on a

stratified collisionless plasma [9].

After all the discussion of the complicated derivation of the ponderomotive

force by Landau and Lifshitz’s stress tensor method, or by Schluter’s two-fluid

method, this author notes that Eqn (5.12) is highly suggestive of a far simpler

method of deriving the nonlinear ponderomotive force density for a plasma in

an electromagnetic field. This simple method takes advantage of the fact that

it is already known that the Maxwell stress tensor U and time derivative of the

Poynting vector can be derived in the ‘canonical’ way from the Lagrangian

L =ε02E2 − 1

2µ0

B2,

as was discussed in §2.2. Presented with a problem in plasma physics, one could

be motivated to make a simple modification of the above Lagrangian by replacing

ε0 with ε0εr = ε0η2 to account for the relative permittivity of the plasma. Deriving

the energy-momentum tensor in the standard way and taking its divergence gives

a force density remarkably similar to that in Eqn (5.12):

f = ∇ ·(

U + ε0(η2 − 1)

(E⊗ E− 1

2IE2

))− ε0η2

∂E×B

∂t(5.13)

Only the fact that the refractive index appears as a coefficient for the IE2 and

109

5.3 The Transient Nonlinear Ponderomotive Force

Poynting terms ruins an exact match with Eqn (5.12). However, perhaps Eqn

(5.13) is more general than (5.12), as it would be expected that any alteration of

the permittivity to account for the plasma medium should affect every term in

the force expression that contains the electric field.

Both Eqn (5.12) and (5.13) reduce to the divergence of the Maxwell stress ten-

sor and Poynting time derivative when free space is considered (η = 1). All that

was required to derive Eqn (5.13) was a trivial alteration of the electromagnetic

Lagrangian to account for the change in permittivity of the electric field in the

presence of matter. No mention was made of plasmas and the result is completely

general until choosing to substitute a specific expression for the refractive index,

like one that incorporates plasma physics as in Eqn (1.17) where

η2 = 1−ω2p

ω(1− iν/ω).

The standard electromagnetic Lagrangian can easily be altered to include the

relative permeability of a plasma by substituting µ0µr for µ0 where µr 6= 1. That

such an elementary approach to the problem of laser-plasma physics should yield

results so similar to other (far more complicated) approaches can be viewed as

a validation of both these other approaches and the central tenet of this thesis -

that ‘canonical’ methods applied to complicated physics problems can still yield

interesting results. A canonical approach is highly attractive since all equations

of motion and notions of conserved quantities are exact and completely consistent

with each other, having all arisen from one seminal functional - the Lagrangian.


While Eqn (5.12) accounted for an inhomogeneous electric field (one that was not

constant over all space under consideration), it did not account for any change

in the ponderomotive force over time, as is required to describe the process of

simply switching on and off the laser beam. An additional transient expression

110


for laser-plasma interaction was explored by Zeidler et al. [133] and completed

by Hora in 1985 [12]. The original expression in 3-dimensional vector notation

for the time-dependent force is

f = ∇ ·(

U +

(1 +

1

ω

∂

∂t

)ε0(η

2 − 1)E⊗ E

)− ε0

∂

∂tE×B. (5.14)

Note that the extra terms added to the force expression by Hora are not

Lorentz invariant since it does not take into account the magnetic fields that would

appear outside the rest frame. Also, the argument of momentum conservation

could not be used to validate Eqn (5.14) as was the case with Eqn (5.12). In

an attempt to overcome this, the expression was generalised by Rowlands [11]

[134] to include magnetic fields by expressing the force using the electromagnetic

tensor, such that

f ν = T νµ,µ = Tνµ,µ +∂

∂xµ

((1 +

c

ω

∂

∂t

)ε0(η

2 − 1)

µ20σ

2F ντ

,τFµγ,γ

)(5.15)

where the time derivative of the Poynting vector is absorbed into the complete

4×4 stress-energy tensor Tνµ in the usual way. The conversion of Eqn (5.14) into

an expression that was both gauge and Lorentz invariant was seen as confirmation

of its correctness [18]. However, it is argued in this thesis that (5.15) is not the

correct generalisation of the transient nonlinear ponderomotive force (5.14) for

three separate reasons.

Firstly, the process used by Rowlands to transit from (5.14) to (5.15) involved

taking advantage of Ohm’s Law and Maxwell’s equations whereby

Ei =1

σJ i and − µ0J

µ = F µτ,τ ,

thus substituting a four-vector for the electric field as follows:

Ei → − 1

µ0σF µτ

,τ .

111


In this way, Hora’s expression was neatly converted to four-dimensional no-

tation while also bringing magnetic fields into the fold with the electromagnetic

tensor. The process of substituting Ohm’s Law into Ampere’s Law to describe

plasma physics is reasonable and was used in §1.2.4, §1.2.5 and §5.2 to derive

the dispersion relation. However, if all the relevant equations - Lorentz force,

Maxwell’s equations, Ohm’s Law - are to be physically consistent with the tran-

sient ponderomotive force, then they must all spring from the same source. This

author has already pointed out in §5.2 that while the non-transient ponderomo-

tive force appears to alter the energy-momentum tensor, it (or something very

close to it) can still be derived from the standard electromagnetic Lagrangian

with a simple modification, thus leaving everything else constructed from the La-

grangian (the form Maxwell’s equations) untouched in the process and ensuring

that the whole system is self-consistent. The tensor being differentiated in Eqn

(5.14) is suggestive of a higher derivative theory, a theory that would yield mod-

ifications to Maxwell’s equations at least, and yet the usual equations of motion

are assumed to be valid substitutions in transiting to Eqn (5.15).

While Rowlands was not concerned with higher-derivative Lagrangian theo-

ries, it is clear that in a consistent physical model, if the energy-momentum tensor

differs from the ‘canonical’ expression by the addition of higher-derivatives of the

field, there is no reason to expect the standard Euler-Lagrange equation,

−µ0Jµ = F µτ

,τ , (5.16)

to hold. For example, it has already been shown that in the case of the Podol-

sky Euler-Lagrange equation (4.18), if the Podolsky Lagrangian was modified to

include the standard interaction term JαAα for matter in the presence of the

electromagnetic field, then the Euler-Lagrange equation (4.18) would be

−µ0Jµ = (1 + µ0a

2 ∂2

∂xγxγ)F µτ

,τ .

112


Indeed, the Podolsky Lagrangian presents the biggest challenge to Eqn (5.15).

It was discussed in §4.3 that the Podolsky Lagrangian is the only Lorentz invariant

generalisation of the electromagnetic Lagrangian that yields equations of motion

below sixth-order, yet Eqn (5.15) introduces a term into the tensor that is one

order of derivative higher than anything in the Podolsky tensor (4.21) and only

shares one term (the nontransient term) with the Podolsky tensor out of the four

or five extra terms it contains (depending on whether the tensor in (4.21) or

(4.26) is considered). Since the Podolsky Lagrangian is well known to be the only

Lorentz invariant generalisation of electrodynamics yielding linear equations of

motion below sixth order, it would be expected that any other energy-momentum

tensor expression containing higher derivatives would be some special case of the

Podolsky tensor, but there appears to be no real correlation between the Rowlands

and Podolsky tensors.

Secondly, high temperature, high energy laser-plasma interactions (seen in

ICF, for example) often assume that the plasma is collisionless, in which case the

conductivity is extremely high or approximately infinite. The appearance of the

square of the conductivity in the denominator of the nonlinear ponderomotive

terms of Eqn (5.15) then seems to make them negligible or effectively zero, in

contrast to Hora’s assertion that the nonlinear ponderomotive force dominates

above thermal effects in certain regimes [18].

Thirdly, the transient nonlinear ponderomotive force in Eqn (5.14) is supposed

to account for the time variation in the field itself and it would be expected that

the dominant temporal contribution to the physics would come from the field

and perhaps its derivatives, if not just the field itself. However, only the time

derivative of the derivatives of the field appears in Eqn (5.15).

A final minor point regarding the way Rowlands argued for the gauge and

Lorentz invariance of Eqn (5.15). The expression is of course gauge invariant

since the electromagnetic tensor and its derivatives are gauge invariant. However,

Rowlands mentioned in two papers [11] [134] that the laser frequency ω is a

113


scalar and therefore Lorentz invariant. Of course, the frequency is not invariant

and would be blue- or red-shifted in any frame of reference moving respectively

toward or away from the laser source. The real reason that the appearance of the

laser frequency does not violate Lorentz invariance is the same reason that the

partial derivative with respect to time does not; they appear in the same factor,

1

ω

∂

∂t,

which is dimensionless and therefore invariant under any transformation of space-

time coordinates.

We present here an original alternate method of generalising Eqn (5.14) to four

dimensions while also ensuring gauge and Lorentz invariance. First, consider Eqn

(5.14) with the purely notational changes

Ei → cF i0 → cF µ0.

Note that the components of the electric field can be sent to components of

the electromagnetic tensor, without ambiguity, with a change in the range of the

index i = 1, 2, 3 to µ = 0, 1, 2, 3 since F 00 = 0. This transformation to a four-

vector was what Rowlands did not consider to be possible, and hence he sought

to transform the electric field to a four-vector via the current. Now Eqn (5.14)

looks like

fµ = Tµν,ν +∂

∂xν

((1 +

1

ω

∂

∂t

)η2 − 1

µ0

F µ0F ν0

). (5.17)

The energy-momentum tensor is then simply

T µν = Tµν +

(1 +

1

ω

∂

∂t

)η2 − 1

µ0

F µ0F ν0. (5.18)

If an inclusion of magnetic fields is sought, then perhaps the simplest alteration

would be to ‘allow the magnetic field in’ by changing the indices that are fixed

114


at 0 to dummy indices α such that

T µν = Tµν +

(1 +

1

ω

∂

∂t

)η2 − 1

µ0

F µαF να. (5.19)

When α = 0, there is no change of sign in lowering the index where F ν0 → F να,

so Eqn (5.19) is consistent with (5.18). When α 6= 0, it is true that F να =

−F να, but since the original expression did not include these terms, the change

of sign can be considered part of the generalisation. Now the ponderomotive

term in (5.19) again appears as a ‘refractive’ modification to the Maxwell energy-

momentum tensor Tµν , which contains the term

1

µ0

F µαF να.

This is analogous to the three dimensional result in Eqn (5.13), although

the time derivative of the ponderomotive term in (5.19) still escapes a canonical

explanation as it is one order of derivative lower than any of the Podolsky terms

in (4.21). However, (5.19) is Lorentz and gauge invariant.

Compare (5.19) to the Rowlands expression,

T νµ = Tνµ +

(1 +

c

ω

∂

∂x0

)ε0(η

2 − 1)

µ20σ

2F ντ

,τFµγ,γ. (5.20)

The difference between (5.19) and (5.20) is of course the higher derivatives

and squared conductivity term in the denominator of Rowlands’ expression. Is

(5.19) any more physically meaningful as a generalisation of the tensor in Eqn

(5.14)? This question probably requires experimental verification in laser-plasma

experiments at the kind of ultra high laser intensities where the ponderomotive

force becomes dominant. However, the advantage of (5.19) over the Rowlands

expression is that it did not require any assumptions about the field equations to

derive it, merely a change in notation and a ‘generalisation’ in letting the indices

of the electromagnetic tensor roam across their full range. The new expression

115


(5.19) also contains time derivatives of the field and not the field derivatives,

which seems more physically sensible and in line with the original Eqn (5.14).

The tensor in (5.19) is also, like the Rowlands expression, both gauge and Lorentz

invariant but still appears to lack any canonical way of deriving the transient term

from a Lagrangian.

It is this author’s opinion that the transient nonlinear ponderomotive force

derived from (5.19),

fµ = Tµν,ν +∂

∂xν

((1 +

1

ω

∂

∂t

)η2 − 1

µ0

F µαF να

), (5.21)

is the correct generalisation of Eqn (5.14), or at least it is the gauge and Lorentz

invariant version that most closely approximates Eqn (5.14).

116

6

Concluding Remarks

Closing the gap between theory and experiment in ICF schemes is likely going

to take many more years, or even decades. Every time a new and more powerful

laser system has been built to achieve ignition in ICF, the additional energy of

the laser system served to magnify other physical complexities and negate the

simulations based on earlier systems. It is currently not clear whether ICF will

ever be useful on a commercial scale. However, hope still remains in the fact

that physicists have not reached a dead end in what they do not know. Consider

the particle physicists at the Large Hadron Collider (LHC) who, having recently

found the Higgs boson exactly as they expected to, are left with no obvious leads

to follow (it was hoped that some deviation from predictions would light the way

to new physics.) The situation was elegantly summed up in a recent article which

stated that “NIF physicists wish their simulations were better; LHC physicists

complain that theirs were too good” [4].

The extreme condensed matter states encountered ICF are not well under-

stood and it is likely that all the principles and assumptions used in the past

are no longer good enough. Nature is always described by the laws of quantum

mechanics and relativity (and some unknown laws in between), whether or not we

can practically ignore them in everyday pursuits. The question of how much can

still be practically ignored in ultra-high intensity laser-plasma physics remains

open. Even taking into account the magnitude of complexity of these problems,

117

it may be that new physics is required to accurately describe laser-plasma inter-

action at high energies. Some have suggested that it is even possible to probe the

physics of particle pair production at energy levels currently available [135].

With these problems in mind, the main thesis put forward in the preced-

ing chapters is that analytical solutions to problems formulated in terms of La-

grangian or Hamiltonian physics can yield interesting results, even when applied

to fields as fraught with complexity as plasma physics. Much of the literature

reviewed herein has focused on reproducing equations of motion that are already

well-known to find some added level of insight through the canonical structure,

if not for purely aesthetic reasons.

This author submits that it is far more interesting to guess a new Lagrangian

or Hamiltonian for a given problem and then derive all the equations of motion

and conservation laws via Hamilton’s principle. The results may not coincide

with other expressions, indeed, it would be hoped that they do not if any new

insight is to be gained. All the salient features of the problem can be uncovered

provided the choice of Lagrangian functional is as reasonable a guess at the true

nature of the physics as possible.

It was this idea that led this author to propose in Chapter 4 that the Podolsky

Lagrangian could be applied to the case of laser-plasma interaction. The Podol-

sky Lagrangian represents a simple modification to Maxwell’s electrodynamics

that eliminates the infinite self-energy of the electron and incorporates higher

frequencies that are usually neglected in a first approximation. If the Podolsky

Lagrangian represents the true nature of electromagnetism with a new constant of

nature determining physics below the femtometre scale, then it is most definitely

applicable to the extreme condensed matter states encountered in ICF. However,

if the Podolsky Lagrangian does not represent the true physics of electromag-

netism, then perhaps its form could still be used to describe plasma physics at a

certain scale, specifically, that of the Debye length in a plasma. In either case,

it is this author’s opinion that the results presented in Chapter 4, especially the

118

new plasma dispersion relation Eqn (4.27), will be of use to physicists studying

laser-plasma interaction now and in the future.

As a side note to the general theme in Chapter 4, it is worth noting that this

author’s derivation of a gauge invariant energy-momentum tensor for any gauge

invariant Lagrangian dependent on second order derivatives of the coordinates is

fully general. Even if a particular Lagrangian is not gauge invariant, any gauge

invariant terms can be separated and the corresponding tensor found with ease

according to Eqn (4.15), which may simplify calculations considerably.

In Chapter 5, the case of the nonlinear ponderomotive force in laser-plasma

interactions was addressed. After reviewing a derivation of the nonlinear pondero-

motive force based on the two-fluid plasma model, and another derivation based

on a stress tensor method due to Landau & Lifshitz and Hora, it was pointed out

by this author that something very close to this force expression could be derived

via Hamilton’s principle. All that was required for this canonical derivation of

the force was a modification of the coefficient of the squared electric field mag-

nitude in the standard electromagnetic Lagrangian to account for the fact that

it describes a laser beam interacting with a dielectric (plasma). This method

was extended in a critical review by this author of the work by Rowlands, who

had derived what he considered to be the Lorentz invariant expression of the

transient nonlinear ponderomotive force in covariant notation. However, it was

pointed out by this author that a simpler generalisation of the transient nonlinear

ponderomotive force was possible, and that this generalisation was more in line

with the idea that the force is a special case of the usual canonical expression,

albeit modified for a plasma.

The results of this thesis have therefore flown entirely from the realm of vari-

ational principles applied to plasma physics. In one sense, it is no surprise to

experienced physicists that the equations of motion for any system should be de-

rived so easily from Hamilton’s principle, given an appropriate Lagrangian. On

the other hand, it seems quite surprising that such results should be found for

119

the fiendishly complicated field of laser-plasma interaction. Whether or not the

original material presented in this thesis will improve the accuracy of theoretical

models of laser-plasma interaction is a question that can only be answered by

experiments. However, the central tenet of this thesis remains that Hamilton’s

principle is the most powerful tool at the disposal of any theoretical physicist,

and that physicists should not be daunted by applying it to ‘real-world’ problems

that appear to be too complicated for it to handle.

120

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(1952 to 1962). Technical report, Lawrence Livermore National Lab.,

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[2] J. H. Nuckolls, L. Wood, A. Thiessen, and G. Zimmerman. Laser

Compression of Matter to Super-High Densities: Thermonuclear

(CTR) Applications. Nature, 239(15):139–142, 1972. 1

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