PHYS3001Classical MechanicsRobert L. DewarDepartment of
Theoretical PhysicsResearch School of Physical Sciences &
EngineeringThe Australian National UniversityCanberra ACT
[email protected] 1.51May 20, 2001. c (
R.L. Dewar 19982001.iiContents1 Generalized Kinematics 11.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 11.2 Generalized coordinates . . . . . . . . . . . . . . . . . .
. . . 21.3 Example: The ideal uid . . . . . . . . . . . . . . . . .
. . . . 51.4 Variational Calculus . . . . . . . . . . . . . . . . .
. . . . . . 61.4.1 Example: Geodesics . . . . . . . . . . . . . . .
. . . . 91.4.2 Trial function method . . . . . . . . . . . . . . .
. . . 101.5 Constrained variation: Lagrange multipliers . . . . . .
. . . . 101.6 Problems . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 131.6.1 Rigid rod . . . . . . . . . . . . . . . . .
. . . . . . . . 131.6.2 Ecliptic . . . . . . . . . . . . . . . . .
. . . . . . . . . 131.6.3 Curvature of geodesics . . . . . . . . .
. . . . . . . . . 132 Lagrangian Mechanics 152.1 Introduction . . .
. . . . . . . . . . . . . . . . . . . . . . . . . 152.2 Generalized
Newtons 2nd Law . . . . . . . . . . . . . . . . . 162.2.1
Generalized force . . . . . . . . . . . . . . . . . . . . . 162.2.2
Generalized equation of motion . . . . . . . . . . . . . 202.2.3
Example: Motion in Cartesian coordinates . . . . . . . 212.3
Lagranges equations (scalar potential case) . . . . . . . . . .
212.3.1 Hamiltons Principle . . . . . . . . . . . . . . . . . . .
232.4 Lagrangians for some Physical Systems . . . . . . . . . . . .
. 242.4.1 Example 1: 1-D motionthe pendulum . . . . . . . . 242.4.2
Example 2: 2-D motion in a central potential . . . . . 252.4.3
Example 3: 2-D motion with time-varying constraint . 262.4.4
Example 4: Atwoods machine . . . . . . . . . . . . . . 272.4.5
Example 5: Particle in e.m. eld . . . . . . . . . . . . 282.4.6
Example 6: Particle in ideal uid . . . . . . . . . . . . 292.5
Averaged Lagrangian . . . . . . . . . . . . . . . . . . . . . . .
302.5.1 Example: Harmonic oscillator . . . . . . . . . . . . . .
312.6 Transformations of the Lagrangian . . . . . . . . . . . . . .
. 32iiiiv CONTENTS2.6.1 Point transformations . . . . . . . . . . .
. . . . . . . . 322.6.2 Gauge transformations . . . . . . . . . . .
. . . . . . . 342.7 Symmetries and Noethers theorem . . . . . . . .
. . . . . . . 352.7.1 Time symmetry . . . . . . . . . . . . . . . .
. . . . . . 372.8 Problems . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 372.8.1 Coriolis force and cyclotron motion . .
. . . . . . . . . 372.8.2 Anharmonic oscillator . . . . . . . . . .
. . . . . . . . 383 Hamiltonian Mechanics 413.1 Introduction:
Dynamical systems . . . . . . . . . . . . . . . . 413.2 Mechanics
as a dynamical system . . . . . . . . . . . . . . . . 413.2.1
Lagrangian method . . . . . . . . . . . . . . . . . . . . 413.2.2
Hamiltonian method . . . . . . . . . . . . . . . . . . . 433.2.3
Example 1: Scalar potential . . . . . . . . . . . . . . . 453.2.4
Example 2: Physical pendulum . . . . . . . . . . . . . 473.2.5
Example 3: Motion in e.m. potentials . . . . . . . . . . 483.2.6
Example 4: The generalized N-body system . . . . . . 483.3
Time-Dependent and Autonomous Hamiltonian systems . . . 503.4
Hamiltons Principle in phase space . . . . . . . . . . . . . . .
503.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 533.5.1 Constraints and moving coordinates . . . . . . . .
. . . 533.5.2 Anharmonic oscillator phase space . . . . . . . . . .
. 533.5.3 2-D motion in a magnetic eld . . . . . . . . . . . . . .
534 Canonical transformations 554.1 Introduction . . . . . . . . .
. . . . . . . . . . . . . . . . . . . 554.2 Generating functions .
. . . . . . . . . . . . . . . . . . . . . . 564.2.1 Example 1:
Adiabatic Oscillator . . . . . . . . . . . . . 604.2.2 Example 2:
Point transformations . . . . . . . . . . . . 624.3 Innitesimal
canonical transformations . . . . . . . . . . . . . 634.3.1 Time
evolution . . . . . . . . . . . . . . . . . . . . . . 644.4 Poisson
brackets . . . . . . . . . . . . . . . . . . . . . . . . . .
654.4.1 Symmetries and integrals of motion . . . . . . . . . . .
664.4.2 Perturbation theory . . . . . . . . . . . . . . . . . . . .
664.5 Action-Angle Variables . . . . . . . . . . . . . . . . . . .
. . . 674.6 Properties of canonical transformations . . . . . . . .
. . . . . 694.6.1 Preservation of phase-space volume . . . . . . .
. . . . 694.6.2 Transformation of Poisson brackets . . . . . . . .
. . . 744.7 Problems . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 744.7.1 Coriolis yet again . . . . . . . . . . . . .
. . . . . . . . 744.7.2 Dierence approximations . . . . . . . . . .
. . . . . . 75CONTENTS v5 Answers to Problems 775.1 Chapter 1
Problems . . . . . . . . . . . . . . . . . . . . . . . . 775.2
Chapter 2 Problems . . . . . . . . . . . . . . . . . . . . . . . .
805.3 Chapter 3 Problems . . . . . . . . . . . . . . . . . . . . .
. . . 865.4 Chapter 4 Problems . . . . . . . . . . . . . . . . . .
. . . . . . 936 References and Index 99vi CONTENTSChapter
1Generalized coordinates andvariational principles1.1
IntroductionIn elementary physics courses you were introduced to
the basic ideas of New-tonian mechanics via concrete examples, such
as motion of a particle in agravitational potential, the simple
harmonic oscillator etc. In this course wewill develop a more
abstract viewpoint in which one thinks of the dynamics ofa system
described by an arbitrary number of generalized coordinates, but
inwhich the dynamics can be nonetheless encapsulated in a single
scalar func-tion: the Lagrangian, named after the French
mathematician Joseph LouisLagrange (17361813), or the Hamiltonian,
named after the Irish mathe-matician Sir William Rowan Hamilton
(18051865).This abstract viewpoint is enormously powerful and
underpins quantummechanics and modern nonlinear dynamics. It may or
may not be more ef-cient than elementary approaches for solving
simple problems, but in orderto feel comfortable with the formalism
it is very instructive to do some ele-mentary problems using
abstract methods. Thus we will be revisiting suchexamples as the
harmonic oscillator and the pendulum, but when examplesare set in
this course please remember that you are expected to use the
ap-proaches covered in the course rather than fall back on the
methods youlearnt in First Year.In the following notes the
convention will be used of italicizing the rst useor denition of a
concept. The index can be used to locate these denitionsand the
subsequent occurrences of these words.The present chapter is
essentially geometric. It is concerned with the de-scription of
possible motions of general systems rather than how to
calculate12CHAPTER 1. GENERALIZEDCOORDINATES ANDVARIATIONAL
PRINCIPLESphysical motions from knowledge of forces. Thus we call
the topic generalizedkinematics.1.2 Generalized coordinatesSuppose
we have a system of N particles each moving in 3-space and
in-teracting through arbitrary (nite) forces, then the dynamics of
the totalsystem is described by the motion of a point q qi[i = 1,
2, . . . , 3N =x1, y1, z1, x2, y2, z2, . . . , xN, yN, zN in a
3N-dimensional generalized cong-uration space. The number n = 3N of
generalized coordinates qi is calledthe number of degrees of
freedom. No particular metric is assumede.g.we could equally as
well use spherical polar coordinates (see Fig. 1.1), qi =r1, 1,
1,r2, 2, 2, . . .,rN, N, N, or a more general curvilinear
coordinatesystem.In other systems the generalizedxryzrFigure 1.1:
Position vectorr in Cartesian and SphericalPolar
coordinates.coordinates need not even be spa-tial coordinatese.g.
they couldbe the charges owing in an elec-trical circuit. Thus the
convenientvector-like notation q for the arrayof generalized
coordinates should notbe confused with the notation r forthe
position vector in 3-space. Of-ten the set of generalized
coordi-nates is simply denoted q, but inthese notes we use a bold
font, dis-tinguishing generalized coordinatearrays from 3-vectors
by using a bold slanted font for the former and a boldupright font
for the latter.Vectors are entities independent of which
coordinates are used to repre-sent them, whereas the set of
generalized coordinates changes if we changevariables. For
instance, consider the position vector of a particle in Carte-sian
coordinates x, y, z and in spherical polar coordinates r, , in Fig.
1.1.The vector r represents a point in physical 3-space and thus
does not changewhen we change coordinates,r = xex + yey +zez =
rer(, ) . (1.1)However its representation changes, because of the
change in the unit basisvectors from ex, ey, ez to er, e and ez. On
the other hand, the sets of1.2. GENERALIZED COORDINATES
3generalized coordinates rC x, y, z and rsph r, , z are distinct
enti-ties: they are points in two dierent (though related)
conguration spacesdescribing the particle.q1q2q3 ...
qnq(t)q(t1)q(t2)Figure 1.2: Some possible paths in conguration
space, eachparametrized by the time, t.Sometimes the motion is
constrained to lie within a submanifold of thefull conguration
space.1For instance, we may be interested in the motionof billiard
balls constrained to move within a plane, or particles connectedby
rigid rods. In such cases, where there exists a set of
(functionally inde-pendent) constraint equations, or auxiliary
conditionsfj(q) = 0 , j = 1, 2, , m < n , (1.2)the constraints
are said to be holonomic.Each holonomic constraint reduces the
number of degrees of freedom byone, since it allows us to express
one of the original generalized coordinates as1A manifold is a
mathematical space which can everywhere be described locally by
aCartesian coordinate system, even though extension to a global
nonsingular curvilinearcoordinate system may not be possible (as,
e.g. on a sphere). A manifold can always beregarded as a surface
embedded in a higher dimensional space.4CHAPTER 1.
GENERALIZEDCOORDINATES ANDVARIATIONAL PRINCIPLESa function of the
others and delete it from the set. For instance, for a
particleconstrained to move in a horizontal plane (an idealized
billiard table), thevertical position z = const is a trivial
function of the horizontal coordinatesx and y and the conguration
space becomes two dimensional, q = x, y.Consider the set of all
conceivable paths through conguration space (seeFig. 1.2). Each one
may be parametrized by the time, t: q = q(t). Bydierentiating eq.
(1.2) with respect to time, we nd a set of constraints onthe
generalized velocities, qi dqi/ dt, which we write in dierential
notationasni=1fj(q)qidqi fj(q)q dq = 0 , (1.3)where f/q f/q1, . . .
, f/qn and we use the shorthand dot-productnotationafb ni=1aifbi,
(1.4)where ai and bi are arbitrary conguration space variables.The
condition for functional independence of the m constraints is
thatthere be m nontrivial solutions of eq. (1.3), i.e. that the
rank of the matrixfj(q)/qi be its maximal possible value, m.Note
that not all such dierential constraints lead to holonomic
con-straints. If we are given constraints as a general set of
dierential formsni=1(j)i (q) dqi = 0 , (1.5)then we may or may not
be able to integrate the constraint equations to theform eq. (1.2).
When we can, the forms are said to be complete dierentials.When we
cannot, the constraints are said to be nonholonomic.The latter
case, where we cannot reduce the number of degrees of freedomby the
number of constraints, will not be considered explicitly in these
notes.Furthermore, we shall normally assume that any holonomic
constraints havebeen used to reduce qi[i = 1, . . . , n to a
minimal, unconstrained set. How-ever, we present in Sec. 1.5 an
elegant alternative that may be used whenthis reduction is not
convenient, or is impossible due to the existence ofnonholonomic
constraints.There are situations where there is an innite number of
generalizedcoordinates. For instance, consider a scalar eld (such
as the instaneousamplitude of a wave), (r, t). Here is a
generalized coordinate of thesystem and the position vector r
replaces the index i. Since r is a continuousvariable it ranges
over an innite number of values.1.3. EXAMPLE: THE IDEAL FLUID
5r0x(r0,t)Figure 1.3: A uid element advected from point r = r0 at
time t = 0to r = x(r0, t) at time t.1.3 Example: The ideal uidAs an
example of a system with both an innite number of degrees of
freedomand holonomic constraints, consider a uid with density eld
(r, t), pressureeld p(r, t) and velocity eld v(r, t).Here we are
using the Eulerian description, where the uid quantities ,p and v
are indexed by the actual position, r, at which they take on
theirphysical values at each point in time.However, we can also
index these elds by the initial position, r0, ofthe uid particle
passing through the point r = x(r0, t) at time t (seeFig. 1.3).
This is known as the Lagrangian description. (cf. the
Schrodingerand Heisenberg pictures in quantum mechanics.) We shall
denote elds inthe Lagrangian description by use of a subscript L:
L(r0, t), pL(r0, t) andvL(r0, t) = tx(r0, t).The eld x(r0, t) may
be regarded as an innite set of generalized coor-dinates, the
specication of which gives the state of the uid at time t.
TheJacobian J(r0, t) of the change of coordinates r = x(r0, t) is
dened byJ xx0xy0xz0=
xx0yx0zx0xy0yy0zy0xz0yz0zz0
, (1.6)where x0, y0 and z0 are Cartesian components of r0 and x,
y and z are thecorresponding components of x(r0, t). This gives the
change of volume ofa uid element with initial volume dV0 and nal
volume (at time t) dVthroughdV = J(r0, t) dV0 . (1.7)To see this,
consider dV0 = dx0 dy0 dz0 to be an innitesimal rectangularbox, as
indicated in Fig. 1.3, with sides of length dx0, dy0, dz0. This
uid6CHAPTER 1. GENERALIZEDCOORDINATES ANDVARIATIONAL
PRINCIPLESelement is transformed by the eect of compression and
shear to an innites-imal parallelipiped with sides given by dlx
dx0x/x0, dly dy0x/y0,dlz dz0x/z0. The volume of such a parallepiped
is dlx dlydlz =J dx0 dy0 dz0 2.Are the elds (r0, t) and p(r0, t)
additional generalized coordinates whichneed to be specied at each
point in time? In an ideal uid (i.e. one withno dissipation, also
called an Euler uid) the answer is no, because massconservation, dV
= 0 dV0, allows us to writeL(r0, t) = 0(r0)/J(r0, t) , (1.8)while
the ideal equation of state p( dV )= p0( dV0), where is the ratio
ofspecic heats, givespL(r0, t) = p0(r0)/J(r0, t) , (1.9)where 0 and
p0 are the initial density and pressure elds, respectively.
Theseare, by denition, xed in time, so the only time dependence
occurs throughthe Jacobian J, which we showed in eq. (1.6) to be
completely determinedby the Lagrangian displacement eld x(r0, t).
Thus eqs. (1.8) and (1.9) haveallowed us to reduce the number of
generalized coordinate elds from 5 to3 (the three components of
x)mass conservation and the equation of statehave acted as
holonomic constraints.Note: Mass conservation is valid even for
nonideal uids (provided theyare not reacting and thus changing from
one state to another). However,in a uid with nite dissipation, heat
will be generated by the motion andentropy will be increased in
each uid element, thus invalidating the useof the adiabatic
equation of state. Further, the entropy increase dependson the
complete path of the uid through its state space, not just on
itsinstantaneous state. Thus the pressure cannot be holonomically
constrainedin a nonideal uid.Remark 1.1 A useful model for a hot
plasma is the magnetohydrodynamic(MHD) uidan ideal uid with the
additional property of being a perfectelectrical conductor. This
leads to the magnetic eld B(r, t) being frozen into the plasma, so
that BL also obeys a holonomic constraint in the
Lagrangianrepresentation, but as it is a vector constraint it is a
little too complicated togive here.1.4 Variational CalculusConsider
an objective functional I[q], dened on the space of all
dierentiablepaths between two points in conguration space, q(t1)
and q(t2), as depicted1.4. VARIATIONAL CALCULUS 7in Fig.
1.2I[q]
t2t1dt f(q(t), q(t), t) . (1.10)(As we shall wish to integrate
by parts later, we in fact assume the paths tobe slightly smoother
than simply dierentiable, so that q is also dened.)We suppose our
task is to nd a path that makes I a maximum or min-imum (or at
least stationary) with respect to neighbouring paths. Thus wevary
the path by an amount q(t): q(t) q(t) + q(t). Then the
rstvariation, I, is dened to be the change in I as estimated by
linearizing inq:I[q]
t2t1dtq(t)fq + q(t)f q
. (1.11)Our rst task is to evaluate eq. (1.11) in terms of q(t).
The crucial stephere is the lemma delta and dot commute. That is q
dqdt . (1.12)To prove this, simply go back to denitions: q d(q +q)/
dt dq/ dt =dq/ dt 2.We can now integrate by parts to put I in the
formI[q] =qf q
t2t1+
t2t1dt q(t)fq . (1.13)This consists of an endpoint contribution
and an integral of the variationalderivative f/q, dened byfq fq
ddtf q . (1.14)Remark 1.2 The right-hand side of eq. (1.14) is also
sometimes called thefunctional derivative or Frechet derivative of
I[q]. When using this termi-nology the notation I/q is used instead
of f/q so that we can writeeq. (1.13), for variations qi which
vanish in the neighbourhood of the end-points, asI[q]
t2t1dtni=1qi(t)Iqi(t) ,
q, Iq
, (1.15)8CHAPTER 1. GENERALIZEDCOORDINATES ANDVARIATIONAL
PRINCIPLESwhich may be taken as the most general dening equation
for I/q. Theinner product notation (, ) used above is a kind of
innite-dimensional dotproduct where we not only sum over the index
i, but integrate over the indext. If we recall that the change in
the value of a eld dened on 3-space, e.g.(r), due to an arbitrary
innitesimal change r is = r, which maybe regarded as the denition
of the gradient , we see that the functionalderivative I/q may be
thought of, by analogy, as an innite-dimensionalgradient dened on
the function space of paths.A typical variational problem is to
make I extremal or stationary underarbitrary variations q(t)
holding the endpoints xed. That is, we requireI = 0 functions q(t)
such that q(t1) = q(t2) = 0 . (1.16)Note that this condition does
not necessarily require I to be a minimumor maximumit can be a kind
of saddle point in function space, with someascending and some
descending directions. To determine the nature ofa stationary point
we would need to expand I to second order in qthesecond
variation.In the class of variations in eq.t + qit Figure 1.4: A
time-localizedvariation in generalized coor-dinate qi with support
in therange t to t + .(1.16), the endpoint contribution ineq.
(1.13) vanishes, leaving onlythe contribution of the integral
overt. Since q(t) is arbitrary, we can,in particular, consider
functions witharbitrarily localized support in t, asindicated in
Fig. 1.4. (The supportof a function is just the range overwhich it
is nonzero.) As 0,f/q(t) becomes essentially con-stant over the
support of q in eq. (1.13)and we can move it outside the in-tegral.
Clearly then, I can only bestationary for all such variations if
and only if the variational derivative van-ishes for each value of
t and each index ifq = 0 . (1.17)These n equations are known as the
EulerLagrange equations. Some-times we encounter variational
problems where we wish to extremize I under1.4. VARIATIONAL
CALCULUS 9variations of the endpoints as well, q(t1) = q(t2) = 0.
In such cases wesee from eq. (1.13) that, in addition to eq.
(1.17), stationarity implies thenatural boundary conditionsf q = 0
(1.18)at t1 and t2.1.4.1 Example: GeodesicsIn the above development
we have used the symbol t to denote the indepen-dent variable
because, in applications in dynamics, paths in congurationspace are
naturally parametrized by the time. However, in purely
geometricapplications t is simply an arbitrary label for the
position along a path, andwe shall in this section denote it by to
avoid confusion.The distance along a path is given by integrating
the lengths dl of in-nitesimal line elements, given a metric tensor
gi,j such that( dl)2=ni,j=1dqigi,j dqj . (1.19)In terms of our
parameter , we thus have the length l as a functional of theform
discussed abovel =
21d ni,j=1 qigi,j(q, ) qj1/2. (1.20)A geodesic is a curve
between two points whose length (calculated usingthe given metric)
is stationary against innitesimal variations about thatpath. Thus
the task of nding geodesics ts within the class of
variationalproblems we have discussed, and we can use the
EulerLagrange equationsto nd them. Perhaps the best known result on
geodesics is the fact that theshortest path between two points in a
Euclidean space (one where gi,j = 0for i = j and gi,j = 1 for i =
j) is a straight line. Another well-known resultis that the
shortest path between two points on the surface of a sphere is
agreat circle (see Problem 1.6.3 for a general theorem on geodesics
on a curvedsurface).Geodesics are not necessarily purely
geometrical objects, but can havephysical interpretations. For
instance, suppose we want to nd the shape ofan elastic string
stretched over a slippery surface. The string will adjust itsshape
to minimize its elastic energy. Since the elastic potential energy
is a10CHAPTER 1. GENERALIZEDCOORDINATES ANDVARIATIONAL
PRINCIPLESmonotonically increasing function of the length of the
string, the string willsettle onto a geodesic on the
surface.Geodesics also play an important role in General
Relativity, because theworld line of a photon is a geodesic in
4-dimensional space time, with themetric tensor obeying Einsteins
equations. If the metric is suciently dis-torted, it can happen
that there is not one, but several geodesics betweentwo points, a
fact which explains the phenomenon of gravitational
lensing(multiple images of a distant galaxy behind a closer massive
object).1.4.2 Trial function methodOne advantage of the variational
formulation of a problem is that we canuse trial function methods
to nd approximate solutions. That is, we canmake a clever guess,
q(t) = qK(t, a1, a2, . . . , aK) as to the general form ofthe
solution, using some specic function qK (the trial function)
involving anite number of parameters ak, k = 1, . . . , K. Then we
evaluate the integralin eq. (1.11) (analytically or numerically)
and seek a stationary point ofthe resulting function I(a1, a2, . .
. , aK) in the K-dimensional space of theparameters ak. Varying the
ak the variation in I isI =Kk=1Iakak . (1.21)The condition for a
stationary point is thusIak= 0, k = 1, . . . , K , (1.22)that is,
that the K-dimensional gradient of I vanish.Since the true solution
makes the objective functional I stationary withrespect to small
variations, if our guessed trial function solution is close tothe
true solution the error in I will be small. (Of course, because qK
maynot be a reasonable guess for the solution in all ranges of the
parameters,there may be spurious stationary points that must be
rejected because theycannot possibly be close to a true solutionsee
the answer to Problem 2.8.2in Sec. 5.2.)1.5 Constrained variation:
Lagrange multi-pliersAs mentioned in Sec. 1.2 we normally assume
that the holonomic constraintshave been used to reduce the
dimensionality of the conguration space so1.5. CONSTRAINED
VARIATION: LAGRANGE MULTIPLIERS 11that all variations are allowed.
However, it may not be possible to do ananalytic elimination
explicitly. Or it may be that some variables appear ina symmetric
fashion, making it inelegant to eliminate one in favour of
theothers.Thus, even in the holonomic case, it is worth seeking a
method of han-dling constrained variations: when there are one or
more dierential auxiliaryconditions of the form f(j)= 0. In the
nonholonomic case it is mandatoryto consider such variations
because the auxiliary conditions cannot be inte-grated.We denote
the dimension of the conguration space by n. Followingeq. (1.5) we
suppose there are m < n auxiliary conditions of the formf(j)
(j)(q, t)q = 0 . (1.23)The vectors (j), j = 1, . . . , m may be
assumed linearly independent (elsesome of the auxiliary conditions
would be redundant) and thus span an m-dimensional subspace, Vm(t),
of the full n-dimensional linear vector spaceVn occupied by the
unconstrained variations. Thus the equations eq. (1.23)constrain
the variations q to lie within an (n m)-dimensional subspace,Vnm(q,
t), complementary to Vn.The variational problem we seek to solve is
to nd the conditions (thegeneralizations of the EulerLagrange
equations) under which the objectivefunctional I[q] is stationary
with respect to all variations q in Vm(q, t).Apart from this
restriction on the variations, the problem is the same as
thatdescribed by eq. (1.16). The generalization of eq. (1.17) isfqq
= 0 q Vnm(q, t) . (1.24)If there are no constraints, so that m = 0,
then f/q is orthogonal toall vectors in Vn and the only solution is
that f/q 0. Thus eq. (1.24)and eq. (1.17) are equivalent in this
case. However, if m < n, then f/qcan have a nonvanishing
component in the subspace Vm and eq. (1.17) is nolonger valid.An
elegant solution to the problem of generalizing eq. (1.17) was
found byLagrange. Expressed in our linear vector space language,
his idea was thateq. (1.24) can be regarded as the statement that
the projection, (f/q)nm,of f/q into Vnm(q, t) is required to
vanish.However, we can write (f/q)nm as f/q(f/q)m, where (f/q)mis
the projection of f/q into Vm. Now observe that we can write any
vectorin Vm as a linear superposition of the (j)since they form a
basis spanning12CHAPTER 1. GENERALIZEDCOORDINATES ANDVARIATIONAL
PRINCIPLESthis space. Thus we write (f/q)m = j(j), or,
equivalently,
fq
m+mj=1j(j)= 0 , (1.25)where the j(q, t) coecients, as yet to be
determined, are known as theLagrange multipliers. They can be
determined by dotting eq. (1.25) with eachof the m basis vectors
(j), thus providing m equations for the m unknowns.Alternatively,
we can express this variationally as
fq
m+mj=1j(j)q = 0 q Vm(q, t) . (1.26)Since j(j)is the projection
into Vm of f/q, the projection intothe complementary subspace Vnm
is found by subtracting (j(j)) fromf/q. That is,
fq
nm= fq +mj=1j(j)= 0 , (1.27)where the Vnm component of the
second equality follows by eq. (1.24) andthe Vn component from eq.
(1.25). Thus we have n generalized EulerLagrange equations, but
they incorporate the m equations for the, so fararbitrary,
(j)implicit in eq. (1.25). Thus we really only gain (n m)
equa-tions from the variational principle, which is at it should be
because we alsoget m kinematic equations from the constraint
conditionsif we got morefrom the variational principle the problem
would be overdetermined.The variational formulation of the second
equality in eq. (1.27) isfq +mj=1j(j)q = 0 q Vn(q, t) . (1.28)That
is, by using the Lagrange multipliers we have turned the
constrainedvariational problem into an unconstrained one.In the
holonomic case, when the auxiliary conditions are of the form ineq.
(1.2), we may derive eq. (1.28) by unconstrained variation of the
modiedobjective functionalI[q]
t2t1dt (f +mj=1jfj) . (1.29)The auxiliary conditions also follow
from this functional if we require that itbe stationary under
variation of the j.1.6. PROBLEMS 131.6 Problems1.6.1 Rigid rodTwo
particles are connected by a rigid rod so they are constrained to
move axed distance apart. Write down a constraint equation of the
form eq. (1.2)and nd suitable generalized coordinates for the
system incorporating thisholonomic constraint.1.6.2 EclipticSuppose
we know that the angular momentum vectors rkmk rk of a systemof
particles are all nonzero and parallel to the z-axis in a
particular Cartesiancoordinate system. Write down the dierential
constraints implied by thisfact, and show that they lead to a set
of holonomic constraints. Hence writedown suitable generalized
coordinates for the system.1.6.3 Curvature of geodesicsShow that
any geodesic r = x() on a two-dimensional manifold S : r =X(, )
embedded in ordinary Euclidean 3-space, where and are arbi-trary
curvilinear coordinates on S, is such that the curvature vector ()
iseverywhere normal to S (or zero).The curvature vector is dened by
de
/ dl, where e
() dx/ dl isthe unit tangent vector at each point along the path
r = x().Hint: First nd f(, , , ) = l, the integrand of the length
functional,l =
f d (which involves nding the metric tensor in , space in terms
ofX/ and X/). Then show that, for any path on S,f = e
X(and similarly for the derivative) andf = e
ddX ,and again similarly for the derivative.14CHAPTER 1.
GENERALIZEDCOORDINATES ANDVARIATIONAL PRINCIPLESChapter 2Lagrangian
Mechanics2.1 IntroductionThe previous chapter dealt with
generalized kinematicsthe description ofgiven motions in time and
space. In this chapter we deal with one formula-tion (due to
Lagrange) of generalized dynamicsthe derivation of
dierentialequations (equations of motion) for the time evolution of
the generalized co-ordinates. Given appropriate initial conditions,
these (in general, nonlinear)equations of motion specify the motion
uniquely. Thus, in a sense, the mostimportant task of the physicist
is over when the equations of motion havebeen derivedthe rest is
just mathematics or numerical analysis (importantthough these are).
The goal of generalized dynamics is to nd universal formsof the
equations of motion.From elementary mechanics we are all familiar
with Newtons SecondLaw, F = ma for a particle of mass m subjected
to a force F and undergoingan acceleration a r. If we know the
Cartesian components Fi(r, r, t),i = 1, 2, 3, of the force in terms
of the Cartesian coordinates x1 = x, x2 = y,x3 = z and their rst
time derivatives then the equations of motion are theset of three
second-order dierential equations m xiFi = 0.To give a physical
framework for developing our generalized dynamicalformalism we
consider a set of N Newtonian point masses, which may be con-nected
by holonomic constraints so the number n of generalized
coordinatesmay be less than 3N. Indeed, in the case of a rigid body
N is essentiallyinnite, but the number of generalized coordinates
is nite. For example,the generalized coordinates for a rigid body
could be the three Cartesiancoordinates of the centre of mass and
three angles to specify its orientation(known as the Euler angles),
so n = 6 for a rigid body allowed to move freelyin space.1516
CHAPTER 2. LAGRANGIAN MECHANICSWhether the point masses are real
particles like electrons, composite par-ticles like nuclei or
atoms, or mathematical idealizations like the innitesimalvolume
elements in a continuum description, we shall refer to them
generi-cally as particles.Having found a very general form of the
equations of motion (Lagrangesequations), we then nd a variational
principle (Hamiltons Principle) thatgives these equations as its
EulerLagrange equations in the case of no fric-tional dissipation.
This variational principle forms a basis for generalizingeven
beyond Newtonian mechanics (e.g. to dynamics in Special
Relativity).2.2 Generalized Newtons 2nd Law2.2.1 Generalized
forceLet the (constrained) position of each of the N particles
making up the sys-tem be given as a function of the n generalized
coordinates q by rk = xk(q, t),k = 1, . . . , N. If there are
holonomic constraints acting on the particles, thenumber of
generalized coordinates satises the inequality n 3N. Thus,
thesystem may be divided into two subsystemsan exterior subsystem
de-scribed by the n generalized coordinates and an interior
constraint subsys-tem whose (3Nn) coordinates are rigidly related
to the q by the geometricconstraints.In a naive Newtonian approach
we would have to specify the forces act-ing on each particle
(taking into account Newtons Third Law, action andreaction are
equal and opposite), derive the 3N equations of motion foreach
particle and then eliminate all the interior subsystem coordinates
tond the equations of motion of the generalized coordinates only.
This isclearly very inecient, and we already know from elementary
physics thatit is unnecessarywe do not really need an innite number
of equations todescribe the motion of a rigid body. What we seek is
a formulation in whichonly the generalized coordinates, and
generalized forces conjugate to them,appear explicitly. All the
interior coordinates and the forces required to main-tain their
constrained relationships to each other (the forces of
constraint)should be implicit only.To achieve this it turns out to
be fruitful to adopt the viewpoint thatthe total mechanical energy
(or, rather, its change due to the performance ofexternal work, W,
on the system) is the primitive concept, rather than thevector
quantity force. The basic reason is that the energy, a scalar
quantity,needs only specication of the coordinates for its full
description, whereasthe representation of the force, a vector,
depends also on dening a basis2.2. GENERALIZED NEWTONS 2ND LAW
17set on which to resolve it. The choice of basis set is not
obvious when weare using generalized coordinates. [Historically,
force came to be understoodearlier, but energy also has a long
history, see e.g. Ernst Mach The Scienceof Mechanics (Open Court
Publishing, La Salle, Illinois, 1960) pp. 309312,QA802.M14 Hancock.
With the development of Lagrangian and Hamiltonianmethods, and
thermodynamics, energy-based approaches can now be said tobe
dominant in physics.]To illustrate the relation between force and
work, rst consider justone particle. Recall that the work W done on
the particle by a forceF as the particle suers an innesimal
displacement r is W = Fr Fxx+Fyy +Fzz. A single displacement r does
not give enough informa-tion to determine the three components of
F, but if we imagine the thoughtexperiment of displacing the
particle in the three independent directions,r = x ex, y ey and z
ez, determining the work, Wx, Wy, Wz, done ineach case, then we
will have enough equations to deduce the three compo-nents of the
force vector, Fx = Wx/x, Fy = Wy/y, Fz = Wz/z. [IfW can be
integrated to give a function W(r) (which is not always possi-ble),
then we may use standard partial derivative notation: Fx = W/x,Fy =
W/y, Fz = W/z.]The displacements x ex, y ey and z ez are
historically called virtualdisplacements. They are really simply
the same displacements as used inthe mathematical denition of
partial derivatives. Note that the virtualdisplacements are done at
a xed instant in time as if by some invisiblehands, which perform
the work W: this is a thought experimentthedisplacements are
ctitious, not dynamical.Suppose we now transform from the Cartesian
coordinates x, y, z to anarbitrary curvilinear coordinate system
q1, q2, q3(the superscript notationbeing conventional in tensor
calculus). Then an arbitrary virtual displace-ment is given by r
=iqiei, where the basis vectors ei are in general notorthonormal.
The corresponding virtual work is given by W =iqiFi,where Fi eiF.
As in the Cartesian case, this can be used to determine
thegeneralized forces Fi by determining the virtual work done in
three indepen-dent virtual displacements.In tensor calculus the set
Fi is known as the covariant representation ofF, and is in general
distinct from an alternative resolution, the
contravariantrepresentation Fi. (You will meet this terminology
again in relativitytheory.) The energy approach shows that the
covariant, rather than thecontravariant, components of the force
form the natural generalized forcesconjugate to the generalized
coordinates qi.Turning now to the N-particle system as a whole, the
example abovesuggests we dene the set of n generalized forces, Qi,
conjugate to each18 CHAPTER 2. LAGRANGIAN MECHANICSof the n degrees
of freedom qi, to be such that the virtual work done on thesystem
in displacing it by an arbitrary innitesimal amount q at xed timet
is given byW ni=1Qiqi q . (2.1)We now calculate the virtual work in
terms of the displacements of the Nparticles assumed to make up the
system and the forces Fk acting on them.The virtual work isW
=Nk=1Fkrk=ni=1qi
Nk=1Fkxkqi
. (2.2)Comparing eq. (2.1) and eq. (2.2), and noting that they
hold for any q, wecan in particular take all but one of the qi to
be zero to pick out the ithcomponent, giving the generalized force
asQi =Nk=1Fkxkqi. (2.3)When there are holonomic constraints on the
system we decompose theforces acting on the particles into what we
shall call explicit forces and forcesof constraint. (The latter
terminology is standard, but the usage explicitforces seems
newoften they are called applied forces, but this is confus-ing
because they need not originate externally to the system, but also
frominteractions between the particles.)By forces of constraint,
Fcstk , we mean those imposed on the particlesby the rigid rods,
joints, sliding planes etc. that make up the holonomicconstraints
on the system. These forces simply adjust themselves to
whatevervalues are required to maintain the geometric constraint
equations and canbe regarded as private forces that, for most
purposes, we do not need toknow1. Furthermore, we may not be able
to tell what these forces need tobe until we have solved the
equations of motion, so they cannot be assumedknown a priori.The
explicit force on each particle, Fxplk , is the vector sum of any
externallyimposed forces, such as those due to an external
gravitational or electric eld,plus any interaction forces between
particles such as those due to elastic1Of course, in practical
engineering design contexts one should at some stage check thatthe
constraint mechanism is capable of supplying the required force
without deforming orbreaking!2.2. GENERALIZED NEWTONS 2ND LAW
19springs coupling point masses, or to electrostatic attractions
between chargedparticles. If there is friction acting on the
particle, including that due toconstraint mechanisms, then that
must be included in Fxplk as well. Theseare the public forces that
determine the dynamical evolution of the degreesof freedom of the
system and are determined by the conguration q of thesystem at each
instant of time, and perhaps by the generalized velocity q inthe
case of velocity-dependent forces such as those due to friction and
thoseacting on a charged particle moving in a magnetic eld.Figure
2.1 shows a simple system withNmgFFigure 2.1: A body onan inclined
plane as de-scribed in the text.a holonomic constrainta particle
slid-ing on a plane inclined at angle . It issubject to the force
of gravity, mg, thenormal force N, and a friction force F inthe
directions shown. The force of con-straint is N. It does no work
because itis orthogonal to the direction of motion,and its
magnitude is that required to nullout the normal component of the
gravita-tional force, [N[ = mg cos , so as to giveno acceleration
in the normal direction and thus maintain the constraint.We now
make the crucial observation that, because the constraints
areassumed to be provided by rigid, undeformable mechanisms, no
work canbe done on the interior constraint subsystem by the virtual
displacements.That is, no (net) virtual work is done against the
nonfriction forces imposedby the particles on the constraint
mechanisms in performing the variationsq. By Newtons Third Law, the
nonfriction forces acting on the constraintmechanisms are equal and
opposite to the forces of constraint, Fcstk . Thusthe sum over
Fcstk xk vanishes and we can replace the total force Fk withFxplk
in eqs. (2.2) and (2.3). Note: If there are friction forces
associatedwith the constraints there is work done against these,
but this fact does notnegate the above argument because we have
included the friction forces inthe explicit forcesany work done
against friction forces goes into heat whichis dissipated into the
external world, not into mechanical energy within theconstraint
subsystem.That is,Qi =Nk=1Fxplk xkqi. (2.4)For the purpose of
calculating the generalized forces, this is a much morepractical
expression than eq. (2.3) because the Fxplk are known in terms of20
CHAPTER 2. LAGRANGIAN MECHANICSthe instantaneous positions and
velocities. Thus Qi = Qi(q, q, t).2.2.2 Generalized equation of
motionWe now suppose that eq. (2.4) has been used to determine the
Qas functionsof the q (and possible q and t if we have
velocity-dependent forces and time-dependent constraints,
respectively). Then we rewrite eq. (2.3) in the formNk=1Fkxkqi=
Qi(q, q, t) . (2.5)To derive an equation of motion we use Newtons
second law to replaceFk on the left-hand side of eq. (2.5) with
mkrk,Nk=1mkrkxkqiNk=1mk ddt
rkxkqi
rk ddtxkqi
= Qi . (2.6)We now dierentiate rk = xk(q(t), t) with respect to
t to nd the functionvk such that rk = vk(q, q, t): rk = xkt +nj=1
qjxkqj vk(q, q, t) . (2.7)Dierentiating vk wrt qi (treating q and q
as independent variables in partialderivatives) we immediately have
the lemmaxkqi= vk qi. (2.8)The second lemma about partial
derivatives of vk that will be needed isddtxkqi= vkqi, (2.9)which
follows because d/ dt can be replaced by /t + q/q, which com-mutes
with /qi (cf. the interchange of delta and dot lemma in Sec.
1.4).Applying these two lemmas in eq. (2.6) we ndNk=1 ddt
mkvkvk qi
mkvkvkqi
=Nk=1 ddt qi
12mkv2k
qi
12mkv2k
= Qi , (2.10)2.3. LAGRANGES EQUATIONS (SCALAR POTENTIAL CASE)
21In terms of the total kinetic energy of the system, TT
Nk=112mkv2k , (2.11)we write eq. (2.10) compactly as the
generalized Newtons second lawddtT qi Tqi= Qi . (2.12)These n
equations are sometimes called Lagranges equations of motion, butwe
shall reserve this term for a later form [eq. (2.24)] arising when
we assumea special (though very general) form for the Qi. They are
also sometimescalled (e.g. Scheck p. 83) dAlemberts equations, but
this may be historicallyinaccurate so is best avoided.2.2.3
Example: Motion in Cartesian coordinatesLet us check that we can
recover Newtons equations of motion as a specialcase when q = x, y,
z. In this caseT = 12m( x2+ y2+ z2) (2.13)soTx = Ty = Tz = 0
(2.14)andT x = m x , T y = m y , T z = m z . (2.15)Also, from eq.
(2.4) we see that Qi Fi. Substituting in eq. (2.12) weimmediately
recover Newtons 2nd Law in Cartesian formm x = Fx m y = Fy m z = Fy
(2.16)as expected.2.3 Lagranges equations (scalar potential case)In
many problems in physics the forces Fk are derivable from a
potential ,V (r1, r2, , rN). For instance, in the classical N-body
problem the parti-cles are assumed to interact pairwise via a
two-body interaction potential22 CHAPTER 2. LAGRANGIAN
MECHANICSVk,l(rk, rl) Uk,l([rkrl[) such that the force on particle
k due to particle lis given byFk,l = kVk,l= (rkrl)[rkrl[U
k,l(rk,l) , (2.17)where the prime on U denotes the derivative
with respect to its argument,the interparticle distance rk,l [rk
rl[. Then the total force on particle kis found by summing the
forces on it due to all the other particlesFk = l=kkVk,l= kV ,
(2.18)where the N-body potential V is the sum of all distinct
two-body interactionsV Nk=1lkVk,l= 12Nk,l=1
Vk,l . (2.19)In the rst line we counted the interactions once
and only once: notingthat Vk,l = Vl,k, so that the matrix of
interactions is symmetric we havekept only those entries below the
diagonal to avoid double counting. In thesecond, more symmetric,
form we have summed all the o-diagonal entriesof the matrix but
have compensated for the double counting by dividing by2. The
exclusion of the diagonal self-interaction potentials is indicated
byputting a prime on the.Physical examples of such an N-body system
with binary interactionsare: An unmagnetized plasma, where Vk,l is
the Coulomb interactionUk,l(r) = ekel
0r , (2.20)where the ek are the charges on the particles and 0
is the permittivityof free space. We could also allow for the eect
of gravity by addingthe potentialk mkgzk to V , where zk is the
height of the kth particlewith respect to a horizontal reference
plane and mk is its mass.2.3. LAGRANGES EQUATIONS (SCALAR POTENTIAL
CASE) 23 A globular cluster of stars, where Vk,l is the
gravitational interactionUk,l(r) = Gmkmlr , (2.21)where G is the
gravitational constant. A dilute monatomic gas, where Vk,l is the
Van der Waals interaction.However, if the gas is too dense (or
becomes a liquid) we would haveto include 3-body or higher
interactions as the wave functions of morethan two atoms could
overlap simultaneously.Even when the system is subjected to
external forces, such as gravity,and/or holonomic constraints, we
can often still assume that the explicitforces are derivable from a
potentialFxplk = kV . (2.22)Taking into account the constraints, we
see that the potential V (r1, r2, , rN)becomes a function, V (q,
t), in the reduced conguration space. Then, fromeq. (2.4) we haveQi
= Nk=1xkqikV= qiV (q, t) . (2.23)Substituting this form for Qi in
eq. (2.12) we see that the generalizedNewtons equations of motion
can be encapsulated in the very compact form(Lagranges equations of
motion)ddt
L qi
Lqi= 0 , (2.24)where the function L(q, q, t), called the
Lagrangian, is dened asL T V . (2.25)2.3.1 Hamiltons
PrincipleComparing eq. (2.24) with eq. (1.17) we see that Lagranges
equations ofmotion have exactly the same form as the EulerLagrange
equations for thevariational principle S = 0, where the functional
S[q], dened byS
t2t1dt L( q, q, t) , (2.26)24 CHAPTER 2. LAGRANGIAN MECHANICSis
known as the action integral . Since the natural boundary
conditionseq. (1.18) are not physical, the variational principle is
one in which the end-points are to be kept xed.We can now state
Hamiltons Principle: Physical paths in congurationspace are those
for which the action integral is stationary against all
innites-imal variations that keep the endpoints xed.By physical
paths we mean those paths, out of all those that are consistentwith
the constraints, that actually obey the equations of motion with
thegiven Lagrangian.To go beyond the original Newtonian dynamics
with a scalar potentialthat we used to motivate Lagranges
equations, we can instead take Hamil-tons Principle, being such a
simple and geometrically appealing result, as amore fundamental and
natural starting point for Lagrangian dynamics.2.4 Lagrangians for
some Physical Systems2.4.1 Example 1: 1-D motionthe pendulumOne of
the simplest nonlinear systems is( 1 c o s ) l0mgz lFigure 2.2:
Phys-ical pendulum.the one-dimensional physical pendulum (so
calledto distinguish it from the linearized harmonicoscillator
approximation). As depicted in Fig. 2.2,the pendulum consists of a
light rigid rod oflength l, making an angle with the
vertical,swinging from a xed pivot at one end andwith a bob of mass
m attached at the other.The constraint l = const and the
assump-tion of plane motion reduces the system to onedegree of
freedom, described by the general-ized coordinate . (This system is
also calledthe simple pendulum to distinguish it from thespherical
pendulum and compound pendula,which have more than one degree of
freedom.)The potential energy with respect to the equi-librium
position = 0 is V () = mgl(1 cos ), where g is the accelerationdue
to gravity, and the velocity of the bob is v = l, so that the
kineticenergy T = 12mv2 = 12ml2 2. The Lagrangian, T V , is thusL(,
) = 12ml2 2mgl(1 cos ) . (2.27)2.4. LAGRANGIANS FOR SOME PHYSICAL
SYSTEMS 25This is also essentially the Lagrangian for a particle
moving in a sinusoidalspatial potential, so the physical pendulum
provides a paradigm for problemssuch as the motion of an electron
in a crystal lattice or of an ion or electronin a plasma wave.From
eq. (2.27) L/ = ml2 and L/ = mgl sin . Thus, theLagrangian equation
of motion isml2 = mgl sin . (2.28)Expanding the cosine up to
quadratic order in gives the harmonic os-cillator oscillator
approximation (see also Sec. 2.6.2)L Llin 12ml2 2 12mgl2, (2.29)for
which the equation of motion is, dividing through by ml2, + 20 =
0,with 0
g/l.2.4.2 Example 2: 2-D motion in a central potentialLet us
work in plane polar coordinates, q = r, , such thatx = r cos , y =
r sin , (2.30)so that x = r cos r sin , y = r sin +r cos ,
(2.31)whence the kinetic energy T 12( x2+ y2) is found to beT =
12m
r2+r2 2
. (2.32)An alternative derivation of eq. (2.32) may be found by
resolving v into thecomponents rer and re, where er is the unit
vector in the radial directionand e is the unit vector in the
azimuthal direction.We now consider the restricted two body
problemone light particleorbiting about a massive particle which
may be taken to be xed at r = 0(e.g. an electron orbiting about a
proton in the Bohr model of the hydrogenatom, or a planet orbiting
about the sun). Then the potential V = V (r)(given by eq. (2.20) or
eq. (2.21)) is a function only of the radial distancefrom the
central body and not of the angle.26 CHAPTER 2. LAGRANGIAN
MECHANICSThen, from eq. (2.25) the Lagrangian isL = 12m
r2+r2 2
V (r) . (2.33)First we observe that L is independent of (in
which case is said to beignorable). Then L/ 0 and the component of
Lagranges equations,eq. (2.24) becomesddtL = 0 , (2.34)which we may
immediately integrate once to get an integral of the motion,i.e. a
dynamical quantity that is constant along the trajectoryL = const .
(2.35)From eq. (2.33) we see that L/ = mr2 , which is the angular
momentum.Thus eq. (2.33) expresses conservation of angular
momentum.Turning now to the r-componenturFigure 2.3: Planar mo-tion
with a time-varying cen-tripetal constraint as de-scribed in the
text.of Lagranges equations, we see fromeq. (2.33)Lr = mr2V
(r) ,L r = m r , ddtL r = m r .(2.36)From eq. (2.24) we nd the
radialequation of motion to bem r mr2= V
(r) . (2.37)2.4.3 Example 3: 2-D motion with time-varying
con-straintInstead of free motion in a central potential, consider
instead a weight ro-tating about the origin on a frictionless
horizontal surface (see Fig. 2.3) andconstrained by a thread,
initially of length a, that is being pulled steadilydownward at
speed u through a hole at the origin so that the radius r =
aut.Then the Lagrangian is, substituting for r in eq. (2.32),L = T
= 12m
u2+ (a ut)2 2
. (2.38)2.4. LAGRANGIANS FOR SOME PHYSICAL SYSTEMS 27Now only is
an unconstrained generalized coordinate. As before, it isignorable,
and so we again have conservation of angular momentumm(a ut)2 l =
const , (2.39)which equation can be integrated to give as a
function of t, = 0 +(l/mu)[1/(a ut) 1/a] = 0 + lt/[ma(a
ut)].Clearly angular momentum is conserved, because the purely
radial stringcannot exert any torque on the weight. Thus Lagranges
equation of motiongives the correct answer. However, the string is
obviously doing work on thesystem because T = 12[mu2+ (l2/m)/(a
ut)2] is not conserved. Have wenot therefore violated the postulate
in Sec. 2.2.2 of no work being done bythe constraints? The answer
is no because what we assumed in Sec. 2.2.2was that no virtual work
was done by the constraints. The fact that theconstraint is
time-dependent is irrelevant to this postulate, because
virtualdisplacements are done instantaneously at any given
time.2.4.4 Example 4: Atwoods machineConsider two weights of mass
m1 and m2 xx0m1m2Figure 2.4: At-woods machine.suspended from a
frictionless, inertialess pullyof radius a by a rope of xed length,
as de-picted in Fig. 2.4. The height of weight 1 is xwith respect
to the chosen origin and the holo-nomic constraint provided by the
rope allowsus to express the height of weight 2 as x, sothat there
is only one degree of freedom forthis system.The kinetic and
potential energy are T =12(m1 + m2) x2and V = m1gx m2gx. ThusL = T
V is given byL = 12(m1 +m2) x2(m1m2)gx (2.40)and its derivatives
are L/x = (m1m2)gand L/ x = (m1+m2) x, so that the equation of
motion d(L/ x) = L/xbecomes x = m1m2m1 +m2g . (2.41)28 CHAPTER 2.
LAGRANGIAN MECHANICS2.4.5 Example 5: Particle in e.m. eldThe fact
that Lagranges equations are the EulerLagrange equations for
theextraordinarily simple and general Hamiltons Principle (see Sec.
2.3.1) sug-gests that Lagranges equations of motion may have a
wider range of validitythan simply problems where the force is
derivable from a scalar potential.Thus we do not dene L as T V ,
but rather postulate the universal validityof Lagranges equations
of motion (or, equivalently, Hamiltons Principle), fordescribing
non-dissipative classical dynamics and accept any Lagrangian
asvalid that gives the physical equation of motion.In particular,
it is obviously of great physical importance to nd a La-grangian
for which Lagranges equations of motion eq. (2.24) reproduce
theequation of motion of a charged particle in an electromagnetic
eld, underthe inuence of the Lorentz force,mr = eE(r, t) +e rB(r,
t) , (2.42)where e is the charge on the particle of mass m.We
assume the electric and magnetic elds E and B, respectively, tobe
given in terms of the scalar potential and vector potential A by
thestandard relationsE = tA ,B = A . (2.43)The electrostatic
potential energy is e, so we expect part of the La-grangian to be
12m r2 e, but how do we include the vector potential?Clearly we
need to form a scalar since L is a scalar, so we need to dot Awith
one of the naturally occurring vectors in the problem to create a
scalar.The three vectors available are A itself, r and r. However
we do not wish touse A, since AA in the Lagrangian would give an
equation of motion thatis nonlinear in the electromagnetic eld,
contrary to eq. (2.42). Thus we canonly use r and r. Comparing eqs.
(2.42) and (2.43) we see that rA has thesame dimensions as , so let
us try adding that to form the total LagrangianL = 12m r2e + e rA .
(2.44)Taking q q1, q2, q3 = x, y, z we haveLqi= eqi+e3j=1 qjAjqi,
(2.45)2.4. LAGRANGIANS FOR SOME PHYSICAL SYSTEMS 29andddtL qi= m qi
+e dAidt = m qi + eAit +3j=1 qjAiqj . (2.46)Substituting eqs.
(2.45) and (2.46) in eq. (2.24) we ndm qi = eqi Ait
+e3j=1 qjAjqi Aiqj
. (2.47)This is simply eq. (2.42) in Cartesian component form,
so our guessed La-grangian is indeed correct.2.4.6 Example 6:
Particle in ideal uidIn Sec. 1.3 we presented a uid as a system
with an innite number of degreesof freedom. However, if we
concentrate only on the motion of a single uidelement (a test
particle), taking the pressure p and mass density as
known,prescribed functions of r and t the problem becomes only
three-dimensional.Dividing by we write the equation of motion of a
uid element asdvdt = p V , (2.48)where V (r, t) is the potential
energy (usually gravitational) per unit mass.To nd a Lagrangian for
this motion we need to be able to combinethe pressure gradient and
density into an eective potential. In an idealcompressible uid we
have the equation of state p( dV )= p0( dV0), where is the ratio of
specic heats, which we can write asp = const , (2.49)where the
right-hand side is a constant of the motion for the given
testparticle. If we further assume that it is the same constant for
all uidelements in the neighbourhood of the test particle, then we
can take thegradient of the log of eq. (2.49) to get (p)/p = ()/.
Thusp =
p
p2=
p
p .30 CHAPTER 2. LAGRANGIAN MECHANICSSolving for p/ we getp = h
(2.50)where the enthalpy (per unit mass) is dened byh 1p .
(2.51)Using eq. (2.50) in eq. (2.48) we recognize it as the
equation of motionfor a particle of unit mass with total potential
energy h + V . Thus theLagrangian isL = 12 r2h V . (2.52)2.5
Averaged LagrangianFrom Sec. 2.3.1 we know that Lagrangian dynamics
has a variational for-mulation, and so we expect that trial
function methods (see Sec. 1.4.2) maybe useful as a way of
generating approximate solutions of the Lagrangianequations of
motion. In particular, suppose we know that the solutions
areoscillatory functions of t with a frequency much higher than the
inverse ofany characteristic time for slow changes in the
parameters of the system (thechanges being then said to occur
adiabatically). Then we may use a trialfunction of the formq(t) =
q((t), A1(t), A2(t), . . .) (2.53)where q is a 2-periodic function
of , the phase of the rapid oscillations, andthe Ak are a set of
slowly varying amplitudes characterizing the waveform(e.g. see
Problem 2.8.2). Thus, q = (t) q + A1 qA1+ A2 qA2+ , (2.54)where the
instantaneous frequency is dened by(t) (t) . (2.55)Since d ln Ak/
dt, to a rst approximation we may keep only the rstterm in eq.
(2.54). Thus our approximate L is a function of , but not ofA1, A2
etc.Now take the time integration in the action integral to be over
a timelong compared with the period of oscillation, but short
compared with the2.5. AVERAGED LAGRANGIAN 31timescale for changes
in the system parameters. Thus only the phase-averageof L,L
20d2 L (2.56)contributes to the action,S
t2t1dt L(, A1(t), A2(t), . . .) (2.57)for the class of
oscillatory physical solutions we seek. Note that the averagingin
eq. (2.56) removes all direct dependence on , so L depends only on
itstime derivative .Thus, we have a new, approximate form of
Hamiltons Principle in whichthe averaged Lagrangian replaces the
exact Lagrangian, and in which theset , A1, A2, . . . replaces the
set q1, q2, . . . as the generalized variables.Requiring S to be
stationary within the class of quasiperiodic trial functionsthen
gives the new adiabatic EulerLagrange equationsddtL = 0 ,
(2.58)andLAk= 0 , k = 1, 2, . . . . (2.59)The rst equation
expresses the conservation of the adiabatic invariant,J = const,
where J is known as the oscillator actionJ L . (2.60)The second set
of equations gives relations between and the Aks that givethe
waveform and the instantaneous frequency.2.5.1 Example: Harmonic
oscillatorConsider a weight of mass m at the end of a light spring
with spring constantk = m20. Then the kinetic energy is T = 12m
x2and the potential energy isV = 12kx2= 12m20x2. Thus the natural
form of the Lagrangian, T V , isL = 12m( x220x2) . (2.61)Now
consider that the length of the spring, or perhaps the spring
constant,changes slowly with time, so that 0 = 0(t), with d ln 0/
dt 0).(c) Consider a charged particle constrained to move on a
non-rotating smoothinsulating sphere, immersed in a uniform
magnetic eld B = Bez, on whichthe electrostatic potential is a
function of latitude and longitude. Write downthe Lagrangian in the
same generalized coordinates as above and show it isthe same as
that for the particle on the rotating planet with
appropriateidentications of and V .2.8.2 Anharmonic
oscillatorConsider the following potential V , corresponding to a
particle of mass moscillating along the x-axis under the inuence of
a nonideal spring (i.e. one2.8. PROBLEMS 39with a nonlinear
restoring force),V (x) = m202
x2+ x4l20
,where the constant 0 is the angular frequency of oscillations
having ampli-tude small compared with the characteristic length l0,
and = 1 dependson whether the spring is soft ( = 1) or hard ( =
+1).Consider the trial functionx = l0 [Acos t +Bcos 3t +C sin 3t]
,where A, B, C are the nondimensionalized amplitudes of the
fundamentaland third harmonic, respectively, and is the nonlinearly
shifted frequency.By using this trial function in the time-averaged
Hamiltons Principle, ndimplicit relations giving approximate
expressions for , B and C as functionsof A. Show that C 0. The
trial function is strictly appropriate only to thecase A < 1,
but plot /0 and B vs. A from 0 to 1 in the case of both ahard and a
soft spring. (You are encouraged to use Maple or Mathematicaand/or
MatLab in this problem.)40 CHAPTER 2. LAGRANGIAN MECHANICSChapter
3Hamiltonian Mechanics3.1 Introduction: Dynamical
systemsMathematically, a continuous-time dynamical system is dened
to be a sys-tem of rst order dierential equations z = f(z, t) , t R
, (3.1)where f is known as the vector eld and R is the set of real
numbers. Thespace in which the set of time-dependent variables z is
dened is called phasespace.Sometimes we also talk about a
discrete-time dynamical system. This isa recursion relation,
dierence equation or iterated mapzt+1 = f(z, t) , t Z , (3.2)where
f is known as the map (or mapping) and Z is the set of all
integers. . . , 2, 1, 0, 1, 2, . . ..Typically, such systems
exhibit long-time phenomena like attracting andrepelling xed points
and limit cycles, or more complex structures such asstrange
attractors. In this chapter we show how to reformulate
nondissipativeLagrangian mechanics as a dynamical system, but shall
nd that it is a veryspecial case where the above-mentioned
phenomena cannot occur.3.2 Mechanics as a dynamical system3.2.1
Lagrangian methodLagranges equations do not form a dynamical
system, because they implic-itly contain second-order derivatives,
q. However, there is a standard way to4142 CHAPTER 3. HAMILTONIAN
MECHANICSobtain a system of rst-order equations from a second-order
system, whichis to double the size of the space of time-dependent
variables by treatingthe generalized velocities u as independent of
the generalized coordinates, sothat the dynamical system is q = u,
u = q(q, u, t). Then the phase spaceis of dimension 2n. This trick
is used very frequently in numerical problems,because the standard
numerical integrators require the problem to be posedin terms of
systems of rst-order dierential equations.In the particular case of
Lagrangian mechanics, we have from eq. (2.24),expanding out the
total derivative using the chain rule and moving all butthe
highest-order time derivatives to the right-hand side,nj=12L qi qj
qj = Lqi 2L qit nj=12L qiqj qj . (3.3)The matrix H acting on q,
whose elements are given byHi,j 2L qi qj, (3.4)is called the
Hessian matrix. It is a kind of generalized mass tensor (seeSec.
3.2.6), and for our method to work we require it to be nonsingular,
sothat its inverse, H1, exists and we can nd q. Then our dynamical
systembecomes q = u , u = H1Lq 2L qt 2L qq q
. (3.5)Remark 3.1 The Lagrangian method per se does not break
down if the Hes-sian is singular, only our attempt to force it into
the standard dynamicalsystem framework. We can still formally solve
the dynamics in the follow-ing manner: Suppose H is singular, with
rank n m. Then, within the n-dimensional linear vector space ^ on
which H acts, there is an m-dimensionalsubspace (the nullspace)
such that H q 0 for all q . We can solveeq. (3.3) for the component
of q lying in the complementary subspace ^ `provided the right-hand
side satises the solubility condition that it have nocomponent in .
The component of q lying in cannot be found directly,but the
solubility condition provides m constraints that complete the
deter-mination of the dynamics.As a simple example, suppose L does
not depend on one of the generalizedvelocities, qi. Then L/ qi 0
and the ith component of the Lagrangeequations of motion eq. (2.24)
reduces to the constraint L/qi = 0. 23.2. MECHANICS AS A DYNAMICAL
SYSTEM 433.2.2 Hamiltonian methodWe can achieve our aim of nding 2n
rst-order dierential equations byusing many choices of auxiliary
variables other than u. These will be morecomplicated functions of
the generalized velocities, but the extra freedom ofchoice may also
bring advantages.In particular, Hamilton realised that it is very
natural to use as the newauxiliary variables the set p = pi[i = 1,
. . . , n dened bypi qiL(q, q, t) , (3.6)where pi is called the
generalized momentum canonically conjugate to qi.We shall always
assume that eq. (3.6) can be solved to give q as a functionof q and
p q = u(q, p, t) . (3.7)Remark 3.2 We have in eect changed
variables from u to p, and such achange of variables can only be
performed if the Jacobian matrix pi/uj =2L/ qi qj is nonsingular.
From eq. (3.4) we recognize this matrix as beingthe Hessian we
encountered in the Lagrangian approach to constructing adynamical
system. Thus in either approach we require the Hessian to
benonsingular (i.e. for its determinant to be nonzero). This
condition is usuallytrivially satised, but there are physical
problems (e.g. if the Lagrangian doesnot depend on one of the
generalized velocities) when this is not the case.However, in the
standard Hamiltonian theory covered in this course it isalways
assumed to hold.The reason for dening p as in eq. (3.6) is that L/
q occurs explicitlyin Lagranges equations, eq. (2.24), so we
immediately get an equation ofmotion for p p = L(q, q, t)q
q=u(q,p,t). (3.8)Equations (3.7) and (3.8) do indeed form a
dynamical system, but so farit looks rather unsatisfactory: now u
is dened only implicitly as a functionof the phase-space variables
q and p, yet the right-hand side of eq. (3.8)involves a partial
derivative in which the q-dependence of u is ignored!We can x the
latter problem by holding p xed in partial derivativeswith respect
to q (because it is an independent phase-space variable) butthen
subtracting a correction term to cancel the contribution coming
from44 CHAPTER 3. HAMILTONIAN MECHANICSthe q-dependence of u.
Applying the chain rule and then using eqs. (3.6)and (3.7) we get p
= L(q, u, t)q ni=1Luiuiq= L(q, u, t)q ni=1piuiq= q[L(q, u, t) pu]
Hq , (3.9)where we have dened a new function to replace the
Lagrangian, namely theHamiltonianH(q, p, t) pu L(q, u, t) .
(3.10)Given the importance of H/q it is natural to investigate
whetherH/p plays a signicant role as well. Dierentiating eq. (3.10)
we getHp = u(q, p, t) +ni=1pi uiL(q, u, t)
uip= q , (3.11)where the vanishing of the expression in the
square bracket and the identi-cation of u with q follows from eqs.
(3.6) and (3.7).Summarizing eqs. (3.9) and (3.11), q = Hp p = Hq ,
(3.12)These equations are known as Hamiltons equations of motion.
As withLagranges equations they express the dynamics of a system
with an arbitrarynumber of degrees of freedom in terms of a single
scalar function! Unlike theLagrangian dynamical system, the
phase-space variables are treated on acompletely even footingin
Hamiltonian mechanics both the conguration-space variables q and
momentum-space variables p are generalized coordi-nates. We dene
canonical coordinates as phase-space coordinates such thatthe
equations of motion can be expressed in the form of eq. (3.12) and
acanonical system as one for which canonical coordinates exist.3.2.
MECHANICS AS A DYNAMICAL SYSTEM 45Remark 3.3 The transition from
the Lagrangian to the Hamiltonian in or-der to handle the changed
meaning of partial derivatives after a change ofvariable is a
special case of a technique known as a Legendre transformation.It
is encountered quite often in physics and physical chemistry,
especially inthermodynamics.3.2.3 Example 1: Scalar
potentialConsider the Lagrangian for a particle in Cartesian
coordinates, so q =x, y, z may be replaced by r = xex +yey +zez.
Also assume that it movesunder the inuence of a scalar potential V
(r, t) so that the natural form ofthe Lagrangian isL = T V = 12m[
r[2V (r, t) . (3.13)Then, from eq. (3.6)p p L r = m r , (3.14)so
that in this case the canonical momentum is the same as the
ordinarykinematic momentum. Equation (3.14) is solved trivially to
give q = u(p)where u(p) = p/m. Thus, from eq. (3.10) we haveH =
[p[2m
[p[22m V (r, t)
= [p[22m +V (r, t)= T +V . (3.15)That is, the Hamiltonian is
equal to the total energy of the system, ki-netic plus potential.
The fact that the Hamiltonian is an important physicalquantity,
whereas the physical meaning of the Lagrangian is more obscure,is
one of the appealing features of the Hamiltonian approach. Both
theLagrangian and Hamiltonian have the dimensions of energy, and
both ap-proaches can be called energy methods. They are
characterized by the useof scalar quantities rather than the
vectors encountered in the direct use ofNewtons second law. This
has both the theoretical advantage of leading tovery general
formulations of mechanics and the practical benet of avoidingsome
vector manipulations when changing between coordinate systems
(infact, Lagrangian and Hamiltonian methods were developed before
modernvector notation was invented).46 CHAPTER 3. HAMILTONIAN
MECHANICSHarmonic OscillatorAn example is the harmonic oscillator
Hamiltonian corresponding to theLagrangian, eq. (2.61)H = p22m +
m0x22 . (3.16)From eq. (3.12) the Hamiltonian equations of motion
are x = pm p = m0x . (3.17)Gauge-transformed Harmonic OscillatorNow
consider the gauge-transformed harmonic oscillator Lagrangian eq.
(2.75)L
= 12m( x2+ 20x x 20x2) .The canonical momentum is thusp = L
/ x = m( x +0x) (3.18)and we see that the gauge transformation
has eected a transformation ofthe canonical momentum, even though
the generalized coordinate x remainsthe same. This is an example of
a canonical transformation, about which wewill have more to say
later.Solving eq. (3.18) for x we nd u(p) = (p m0x)/m. HenceH = p(p
m0x)m
(p m0x)22m +0x(p m0x) 12m20x2
= (p m0x)22m + 12m20x2= T +V . (3.19)Thus, even though L was not
of the natural form T V in this case, theHamiltonian remains equal
to the total energy, thus conrming that it is aquantity with a more
direct physical signifance than the Lagrangian. (Thefunctional form
of the Hamiltonian changes under the gauge transformationhowever,
because the meaning of p changes.)3.2. MECHANICS AS A DYNAMICAL
SYSTEM 473.2.4 Example 2: Physical pendulumA nonlinear
one-dimensional case is provided by the physical pendulum,
in-troduced in Sec. 2.4.1. Using eq. (2.27) in eq. (3.6) we get p =
ml2 . Thus = p/ml2and so the Hamiltonian, H = p L, becomesH(, p) =
p22ml2 + mgl(1 cos ) , (3.20)which again is of the form T +V .By
conservation of energy (see also Sec. 3.3),0 0.2 0.4 0.6 0.8
1-3-2-10123/2p X O OFigure 3.1: Phasespace of the phys-ical
pendulum.the Hamiltonian H = T + V is a constantof the motion so
the nature of the orbits inphase space can be found simply by
plottingthe contours of H as in Fig. 3.1 (which is inunits such
that m = l = g = 1). We see thatthe structure of the phase space is
rather morecomplicated than in the case of the harmonicoscillator
since there are two topologically dis-tinct classes of orbit. One
class is the rotatingorbits for which the pendulum has enough
en-ergy, H > 2mgl, to swing entirely over the topso increases or
decreases secularly (thoughthe physical position x = l sin does
not).The other class is the librating orbits for H < 2mgl
implying that thependulum is trapped in the gravitational potential
well and oscillates like apair of scales (hence the name). For [[
< 1 we may expand the cosine soV mgl2/2 and the system is
approximately a harmonic oscillator. Theequilibrium point = 0 or 2,
labelled O in Fig. 3.1, is a xed point. Theorbits in its
neighbourhood, like those of the harmonic oscillator, remain inthe
neighbourhood for all time (i.e. the xed point is stable or
elliptic).The dividing line H = 2mgl between the two topological
classes of orbit iscalled the separatrix, and on the separatrix
lies another xed point, labelledX in Fig. 3.1. This corresponds to
the case where the pendulum is justbalanced upside down. Almost all
orbits in the neighbourhood of an Xpoint eventually are repelled
far away from it, and thus it is referred to asan unstable or
hyperbolic xed point.48 CHAPTER 3. HAMILTONIAN MECHANICS3.2.5
Example 3: Motion in e.m. potentialsNow consider the case of a
charged particle in an electromagnetic eld withmagnetic vector
potential A and electrostatic potential . The Lagrangianis given by
eq. (2.44) and thus, from eq. (3.6),p L r = m r +eA(r, t) .
(3.21)Thus u(p) = p/meA/m and, from eq. (3.10) we haveH = (p
eA)pm
[p eA[22m + e(p eA)Am e(r, t)
= [p eA(r, t)[22m +e(r, t)= T +V . (3.22)Thus we nd again that,
although the Lagrangian cannot be put into thenatural form T V (r),
the Hamiltonian is still the total energy, kinetic
pluselectrostatic potential energy.3.2.6 Example 4: The generalized
N-body systemLet us now revisit the general case which led to the
original denition of theLagrangian in Sec. 2.3, the case of N
particles interacting via a scalar N-bodypotential, possibly with
constraints (which we here assume to be independentof time), so
that the number of generalized coordinates is n 3N. Then, byeq.
(2.25) and eq. (2.7) (assuming the function xk(q) to be independent
oftime) the natural form of the Lagrangian isL = T V=
12ni,j=1Nk=1mk qixkqixkqj qjV (q)= 12ni,j=1 qii,j qjV (q) ,
(3.23)where the symmetric mass matrixi,j(q)
Nk=1mkxkqixkqj(3.24)3.2. MECHANICS AS A DYNAMICAL SYSTEM 49is the
metric tensor for a conguration-space mass-weighted length
elementds dened by ( ds)2k mk dl2k. From eq. (3.4), we see that it
is the Hessianmatrix for this system.Then, using eq. (3.23) in eq.
(3.6)pi =nj=1i,j(q) qj . (3.25)Assuming none of the particles is
massless, i,j is a positive-denite ma-trix, so its inverse 1i,j
exists and we can formally solve eq. (3.25) for qi togiveui(q, p,
t) =nj=11i,jpj . (3.26)Then, from eq. (3.10) we haveH
=ni,j=1pi1i,jpj
12ni,j,i
,j
=1pi1i,i
i
,j 1j
,jpjV (q)
= 12ni,j=1pi1i,jpj +V (q)= T +V . (3.27)Thus, the Hamiltonian is
again equal to the total energy of the system. Thisresult does not
hold in the case of a time-dependent representation, xk(q, t).[See
Problems 3.5.1(c) and 4.7.1.]Particle in a central potentialAs a
simple, two-dimensional example of a problem in non-Cartesion
coor-dinates we return to the problem of motion in a central
potential, expressedin plane polar coordinates in Sec. 2.4.2.
Recapitulating eq. (2.33),L = 12m
r2+r2 2
V (r) .Comparing with eq. (3.23) we see that the mass matrix is
diagonal = m 00 mr2
, (3.28)and thus can be inverted simply taking the reciprocal of
the diagonal ele-ments. Hence, from eq. (3.27)H = p2r2m + p22mr2 +V
(r) . (3.29)50 CHAPTER 3. HAMILTONIAN MECHANICS3.3 Time-Dependent
and Autonomous Hamil-tonian systemsAn autonomous dynamical system
is one in which the vector eld f [seeeq. (3.1)] depends only on the
phase-space coordinates and has no explicitdependence on t. (The
reason for the name is that explicit time dependencewould come from
external forcing of the system, whereas autonomousmeans independent
or self-governing.) In the Hamiltonian case this meansthat H has no
explicit time dependence. Conversely, if there is an
externaltime-varying perturbation, then H = H(p, q, t).Consider the
time rate of change of the Hamiltonian, H dH/ dt, follow-ing the
phase-space trajectory. Using the Hamiltonian equations of
motion,eq. (3.12),dHdt = Ht + qHq + pHp= Ht + HpHq HqHp Ht .
(3.30)Thus, if H does not depend explicitly on time, so its partial
time deriva-tive vanishes, then its total time derivative also
vanishes. That is, in anautonomous Hamiltonian system the
Hamiltonian is an integral of motion.We have seen above that H can
often be identied as the total energy.When this is the case we may
interpret the completely general result above asa statement of
conservation of energy. More generally, comparing eq. (3.10)and eq.
(2.84) we recognize H as the negative of the energy integral I
forautonomous systems as predicted by Noethers theorem.3.4
Hamiltons Principle in phase spaceLet us express the action
integral S in terms of q and p and show that wemay derive Hamiltons
equations of motion by making S stationary undervariations of the
trajectory in phase space, rather than in conguration space.To nd a
suitable denition for the action integral, we rst rearrangeeq.
(3.10) to get L on the left-hand side. Then we replace u(q, p, t)
by q inthe term pu and thus dene the phase-space LagrangianLph(q,
q, p, t) p q H(q, p, t) , (3.31)3.4. HAMILTONS PRINCIPLE IN PHASE
SPACE 51If q = u(q, p, t) were identically satised, even on
arbitrarily varied phase-space paths, then Lph would simply be L
expressed in phase-space coordi-nates. However, one can easily
construct a counter example to show thatthis is not the case:
consider a variation of the path in which we can varythe direction
of its tangent vector, at some point (q, p), while keeping
thispoint xed (see Sec. 4.2 for an illustration of this). Then q
changes, but uremains the same. Thus Lph and L are the same value
only on the subset ofpaths (which includes the physical paths) for
which p q = pu(q, p, t).Replacing L by Lph in eq. (2.26) we dene
the phase-space action integralSph[q, p] =
t2t1dt Lph(q, p, q, t) . (3.32)We shall now show that requiring
Sph = 0 for arbitrary variations about agiven path implies that
that path is such that Hamiltons equations of motionare satised at
all points along it. Since we concluded above that Lph and Lwere
not the same on arbitrary nonphysical paths, this variational
principleis subtly dierent from the Lagrangian version of Hamiltons
principle, andis sometimes called the modifed Hamiltons
Principle.We know from variational calculus that Sph is stationary
under arbitraryvariations of the phase-space path (with endpoints
xed) if and only if theEulerLagrange equation