Variational Principles and Lagrange’s Equations

Variational Principles

and Lagrange’s Equations

Definitions

• Lagrangian density:

• Lagrangian:

• Action:

• How to find the special value for action corresponding to observable ?

nin

nmi

xxxη ,)(L tzyxx

i

n ,,,?...3,2,1,0

dxdydzL L

t

dttrdL i

mi

,)(

dtt

dttrdLI i

mi

,)(

)(tr Joseph Louis

Lagrange/Giuseppe Luigi

Lagrangia (1736 – 1813)

zyxm rrrr ,,

Variational principle

• Maupertuis: Least Action Principle

• Hamilton: Hamilton’s Variational Principle

• Feynman: Quantum-Mechanical Path Integral Approach

Pierre-Louis Moreau de Maupertuis (1698 – 1759)

Sir William Rowan Hamilton

(1805 – 1865)

Richard Phillips Feynman

(1918 – 1988)

Functionals

• Functional: given any function f(x), produces a number S

• Action is a functional:

• Examples of finding special values of functionals using variational approach:

shortest distance between two points on a plane;the brachistochrone problem;minimum surface of revolution; etc.

)]([ xfS

2

1

,)()]([t

ti

mi

dttdt

trdLtrI

Shortest distance between two points on a plane

• An element of length on a plane is

• Total length of any curve going between points 1 and 2 is

• The condition that the curve is the shortest path is that the functional I takes its minimum value

22 dydxds

2

1

dsI

2

1

2

1 dxdxdy

The brachistochrone problem

• Find a curve joining two points, along which a particle falling from rest under the influence of gravity travels from the highest to the lowest point in the least time

• Brachistochrone solution: the value of the functional t [y(x)] takes its minimum value

dtdsv /

2

112 v

dst

2

1

2

2/1

dxgydxdy

22 dydxds

mgymv

2

2

gyv 2

Calculus of variations

• Consider a functional of the following type

• What function y(x) yields a stationary value (minimum, maximum, or saddle) of J ?

2

1

),...,,,,()]([x

x

dxxyyyyfxyJ dxdyy

x

y

0

),( 22 yx

),( 11 yx


• Assume that function y0(x) yields a stationary value and consider all possible functions in the form:

x

y

0

),( 22 yx

),( 11 yx

...)()()(),( 12

0 xxxyxy

0)()( 21 xx


• In this case our functional becomes a function of α:

• Stationary value condition:

)()(),( 0 xxyxy 0)()( 21 xx

)(),()],([2

1

JdxxfxyJx

x

0)()(

0)(),( 0

ddJ

ddJ

xyxy

Stationary value

2

1

),...,,,()( x

x

dxxyyyfdd

ddJ

2

1

),...,,,(x

x

dxd

xyyydf

2

1

...x

x

dxyyfy

yfy

yf

1

2

3

2

1

.1x

x

dxyyf

2

1

x

x

dxyf

)()(),( 0 xxyxy

Stationary value

2

1

),...,,,()( x

x

dxxyyyfdd

ddJ

2

1

),...,,,(x

x

dxd

xyyydf

2

1

...x

x

dxyyfy

yfy

yf

1

2

3

2

1

.2x

x

dxyyf

2

1

2x

x

dxxy

yf

u

dv

2

1

x

x

yyf

u

v

2

1

x

x

dxyf

dxdy

v

du

)()(),( 0 xxyxy

2

1

x

xyf

2

1

x

x

dxyf

dxd

0)(0)(

2

1

xx

Stationary value

2

1

),...,,,()( x

x

dxxyyyfdd

ddJ

2

1

),...,,,(x

x

dxd

xyyydf

2

1

...x

x

dxyyfy

yfy

yf

1

2

3

2

1

.3x

x

dxyyf

2

1

x

x

yyf

2

1

x

x

dxyf

dxdy

2

1

x

xyf

2

1

x

xyf

dxd

2

1

2

2x

x

dxyf

dxd

Stationary value

2

1

),...,,,()( x

x

dxxyyyfdd

ddJ

2

1

),...,,,(x

x

dxd

xyyydf

2

1

...x

x

dxyyfy

yfy

yf

1

2

3

2

1

x

xyf

2

1

2

2x

x

dxyf

dxd

2

1

x

x

dxyf

2

1

x

x

dxyf

dxd

2

1

...2

2x

x

dxyf

dxd

yf

dxd

yf

2

1

x

xyf

...

...

Stationary value

... ...2

1

2

1

2

2

x

x

x

x yfdx

yf

dxd

yf

dxd

yf

ddJ

),( xyff

2

1

x

x

dxyf

ddJ

0)(

0

ddJ 0

2

1 0 )(),(

x

x xyxy

dxyf

arbitrary0

yf

Trivial …

Stationary value

... ...2

1

2

1

2

2

x

x

x

x yfdx

yf

dxd

yf

dxd

yf

ddJ

),,( xyyff

2

1

x

x

dxyf

dxd

yf

ddJ

0)(

0

ddJ

0 2

1 0

x

x

dxyf

dxd

yf

arbitrary0

yf

dxd

yf

Nontrivial !!!

Shortest distance between two points on a plane

2

1

2

1 dxdxdyI 21 yf

0

yf

dxd

yf

0

10

2

y

ydxd

c

y

y

21

21 c

cy

bxc

cy

21

Straight line!


2

1

2

12 2/1

dxgydxdy

t gyy

f2

1 2

0

yf

dxd

yf

0

1222

123

2

ygy

ydxd

gy

y

Scary!

Recipe

• 1. Bring together structure and fields

• 2. Relate this togetherness to the entire system

• 3. Make them fit best when the fields have observable dependencies:

Structure

Fields Fields

Structure

Physical Laws

Best F

it

mη mη

Back to trajectories and Lagrangians

• How to find the special values for action corresponding to observable trajectories ?

• We look for a stationary action using variational principle

2

1

,)()]([t

ti

mi

dttdt

trdLtrI

)()(),( 0 ttrtr mmm

0)()( 21 tt mm 0)(

0

ddI

2

1

,),()],([)(t

ti

mi

m dttdttrdLtrII

Recipe




Structure

Fields Fields

Structure

Physical Laws

Best F

it

mη mη


• For open systems, we cannot apply variational principle in a consistent way, since integration in not well defined for them

• We look for a stationary action using variational principle for closed systems:

2

1

,),()],([)(t

ti

mi

m dttdttrdLtrII

0)(

0

ddI

dxdydzL L dxdydzdtI L

Stationary value

... ...2

1

2

1

2

2

x

x

x

x yfdx

yf

dxd

yf

dxd

yf

ddJ

),,( xyyff

2

1

x

x

dxyf

dxd

yf

ddJ

0)(

0

ddJ

0 2

1 0

x

x

dxyf

dxd

yf

0

yf

dxd

yf

Nontrivial !!!

Simplest non-trivial case

• Let’s start with the simplest non-trivial result of the variational calculus and see if it can yield observable trajectories

2

1

,),()(t

ti

mi

dttdttrdLI

0)(

0

ddI

2

1

,),(),,(t

t

mm dtt

dttdrtrL

zyxmi

,,1,0

2

1

,,t

tmm dttrrL

Stationary value

... ...2

1

2

1

2

2

x

x

x

x yfdx

yf

dxd

yf

dxd

yf

ddJ

),,( xyyff

2

1

x

x

dxyf

dxd

yf

ddJ

0)(

0

ddJ

0 2

1 0

x

x

dxyf

dxd

yf

0

yf

dxd

yf

Nontrivial !!!

Euler- Lagrange equations

• These equations are called the Euler- Lagrange equations

0)(

0

ddI

2

1

,,t

tmm dttrrLI

0

yf

dxd

yf

0

mm rL

dtd

rL

Joseph Louis Lagrange

(1736 – 1813)

Leonhard Euler (1707 – 1783)

Recipe




Structure

Fields Fields

Structure

Physical Laws

Best F

it

mη mη

How to construct Lagrangians?

• Let us recall some kindergarten stuff

• On our – classical-mechanical – level, we know several types of fundamental interactions:

• Gravitational• Electromagnetic• That’s it

0

mm rL

dtd

rL

Gravitation

• For a particle in a gravitational field, the trajectory is described via 2nd Newton’s Law:

• This system can be approximated as closed

• The structure (symmetry) of the system is described by the gravitational potential

gUdtvmd

)(

gm

),,,( tzyxgg

Sir Isaac Newton(1643 – 1727)

Electromagnetic field

• For a charged particle in an electromagnetic field, the trajectory is described via 2nd Newton’s Law:

• This system can be approximated as closed

• The structure (symmetry) of the system is described by the scalar and vector potentials

),,,(),,,(

tzyxtzyxAA

)( AvqqAqvmdtd

Really???


)( AvqqAqvmdtd

)()( AvqqdtAdq

dtvmd

dtAd

),,,( tzyxAA

zzAy

yAx

xA

tA

AvtA

)(

AvqtAqAvqq

dtvmd

)()()(

AvAvqtAqq

dtvmd

)()()(


AvAvqtAqq

dtvmd

)()()(

FGFGGFGF

)()()(

GF

vAvAAvAv

)()()(

Av

AvqtAqq

dtvmd

)(


• Lorentz force!

AvqtAq

dtvmd

)(

tAE

AB

BvEqdtvmd

)(

Hendrik Lorentz(1853-1928)

Kindergarten

• Thereby:

• In component form

0)( AqvmdtdAvqq

0)()(

dtrmd

rm j

j

g

0)(

dtvmdm g

0)())((

dtqArmd

rArqq jj

j

How to construct Lagrangians?

• Kindergarten stuff:

• The “kindergarten equations” look very similar to the Euler-Lagrange equations! We may be on the right track!

0

jj rL

dtd

rL

0)()(

dtrmd

rm j

j

g

0)())((

dtqArmd

rArqq jj

j

Gravitation

0

jj rL

dtd

rL

0)()(

dtrmd

rm j

j

g

j

g

j rm

rL

)(

j

j

rL

dtd

dtrmd

)(

),,,( trrrTmL zyxg C

rLrmj

j

),,,( trrrS zyx),,,( trrr zyxgg

2)( 222

zyx rrrmL

Gravitation

gzyx mtrrrTL ),,,(

),,,(2

)( 222

trrrSrrrm

L zyxzyx

gzyx mrrrm

L

2

)( 222

Electromagnetism

0

jj rL

dtd

rL

0)())((

dtqArmd

rArqq jj

j

)(2

)( 222

Arqqrrrm

L zyx

Bottom line

• We successfully demonstrated applicability of our recipe

• This approach works not just in classical mechanics only, but in all other fields of physics

Structure

Physical LawsBes

t Fit

Some philosophy

• de Maupertuis on the principle of least action (“Essai de cosmologie”, 1750): “In all the changes that take place in the universe, the sum of the products of each body multiplied by the distance it moves and by the speed with which it moves is the least that is possible.”

• How does an object know in advance what trajectory corresponds to a stationary action???

• Answer: quantum-mechanical pathintegral approach

Pierre-Louis Moreau de Maupertuis (1698 – 1759)

Some philosophy

• Feynman: “Is it true that the particle doesn't just "take the right path" but that it looks at all the other possible trajectories? ... The miracle of it all is, of course, that it does just that. ... It isn't that a particle takes the path of least action but that it smells all the paths in the neighborhood and chooses the one that has the least action ...”


(1918 – 1988)

Some philosophy

• Dyson: “In 1949, Dick Feynman told me about his "sum over histories" version of quantum mechanics. "The electron does anything it likes," he said. "It just goes in any direction at any speed, forward or backward in time, however it likes, and then you add up the amplitudes and it gives you the wave-function." I said to him, "You're crazy." But he wasn't.”

Freeman John Dyson (born 1923)

Some philosophy

• Philosophical meaning of the Lagrangian formalism: structure of a system determines its observable behavior

• So, that's it?

• Why do we need all this?

• In addition to the deep philosophical meaning, Lagrangian formalism offers great many advantages compared to the Newtonian approach

Lagrangian approach: extra goodies

• It is scalar (Newtonian – vectorial)

• Allows introduction of configuration space and efficient description of systems with constrains

• Becomes relatively simpler as the mechanical system becomes more complex

• Applicable outside Newtonian mechanics

• Relates conservation laws with symmetries

• Scale invariance applications

• Gauge invariance applications

Simple example

• Projectile motion gzyx mrrrm

L

2

)( 222

zzyx mgrrrrm

L

2

)( 222

0

yy rL

dtd

rL

0

xx rL

dtd

rL

0

zz rL

dtd

rL

0xrmdtd

0yrmdtd

constrm x

constrm y

zg gr

mgrmdtd

z constgtrz

Another example

• Another Lagrangian

• What is going on?!

xy

zx mgrrm

rrmL 2

2

0

yy rL

dtd

rL

0

xx rL

dtd

rL

0

zz rL

dtd

rL

0 zrmdtd

0yrmdtd

constrm x

constrm y

constgtrz mg

0 0 xrmdtd

Gauge invariance

• For the Lagrangians of the type

• And functions of the type

• Let’s introduce a transformation (gauge transformation):

trrL ii ,,

dt

trdFtrrLtrrL iiiii

,,,,,'

trF i ,

Gauge invariance

dtdFLL ' j

j j

rrF

tF

dtdF

trFF i ,

dtdF

rrL

rL

iii

'

jj jii

rrF

tF

rrL

jj jiii

rrrF

trF

rL

22

Gauge invariance

dtdFLL ' j

j j

rrF

tF

dtdF

trFF i ,

dtdF

rrL

rL

iii '

jj jii

rrF

tF

rrL

ii rF

rL

iii rF

dtd

rL

dtd

rL

dtd

'

ii rF

trL

dtd

jj ij

rrF

r

Gauge invariance

jj ijiii

rrF

rrF

trL

dtd

rL

dtd

'

jj ijii

rrrF

rtF

rL

dtd

22

jj jiiii

rrrF

trF

rL

rL

22'

ii rL

rL

dtd

'' ii r

LrL

dtd

0

Back to the question: How to construct Lagrangians?

• Ambiguity: different Lagrangians result in the same equations of motion

• How to select a Lagrangian appropriately?

• It is a matter of taste and art

• It is a question of symmetries of the physical system one wishes to describe

• Conventionally, and for expediency, for most applications in classical mechanics:

VTL

Cylindrically symmetric potential

• Motion in a potential that depends only on the distance to the z axis

• It is convenient to work in cylindrical coordinates

• Then

22

222

2)(

yxzyx rrVrrrm

L

zrrrrr zyx ;sin ;cos

zr

rrr

rrr

z

y

x

cossin

sincos


• How to rewrite the equations of motion in cylindrical coordinates?

22

222

2)(

yxzyx rrVrrrm

L

22222

sincos2

rrVzm

2

)cossin( 2 rrm

2)sincos( 2 rrm

)(2

)( 2222

rVzrrm

0

jj rL

dtd

rL

Generalized coordinates

• Instead of re-deriving the Euler-Lagrange equations explicitly for each problem (e.g. cylindrical coordinates), we introduce a concept of generalized coordinates

• Let us consider a set of coordinates

• Assume that the Euler-Lagrange equations hold for these variables

• Consider a new set of (generalized) coordinates

),...,,(: 21 Ni rrrr

0

ii rL

dtd

rL

),,...,,( 21 trrrqq Njj


• We can, in theory, invert these equations:

• Let us do some calculations:

N

i m

i

im qr

rL

qL

1

),,...,,( 21 trrrqq Nmm

),,...,,( 21 tqqqrr Mii

M

mm

m

iii q

qr

trr

1

N

i m

i

im qr

rL

dtd

qL

dtd

1

N

i m

i

i qr

rL

dtd

1

m

iN

i i

N

i m

i

i qr

dtd

rL

qr

rL

dtd

11

m

i

m

i

qr

qr


• The Euler-Lagrange equations are the same in generalized coordinates!!!

M

kk

m

i

k

i

m

i

m

i qqr

qr

qr

tqr

dtd

1

m

iN

i i

N

i m

i

im qr

dtd

rL

qr

rL

dtd

qL

dtd

11

M

kk

k

ii

m

qqr

tr

q 1

M

mm

m

iii q

qr

trr

1

m

i

qr

mqL

dtd

m

iN

i i qr

rL

1 mq

L

N

i m

i

i qr

rL

1


• If the Euler-Lagrange equations are true for one set of coordinates, then they are also true for the other set

ii rL

rL

dtd

),,...,,( 21 trrrqq Nmm

mm qL

qL

dtd


• Radial force causes a change in radial momentum and a centripetal acceleration

)(2

)( 2222

rVzrrmL

0

jj qL

dtd

qL

),,(: zrqi 0

rL

dtd

rL

rrV

)( 0)(

dtrmd

dtrmdmr

rrV )()( 2

2mr


• Angular momentum relative to the z axis is a constant

)(2

)( 2222

rVzrrmL

0

jj qL

dtd

qL

),,(: zrqi 0

L

dtdL

0 0)( 2

dtmrd

constmrrmr )(2

0)( 2

dtmrd


• Axial component of velocity does not change

)(2

)( 2222

rVzrrmL

0

jj qL

dtd

qL

),,(: zrqi 0

zL

dtd

zL

0 0)(

dtzmd

constzm

0)(

dtzmd

Symmetries and conservation laws

• The most beautiful and useful illustration of the “structure vs observed behavior” philosophy is the link between symmetries and conservation laws

• Conjugate momentum for coordinate :

• If Lagrangian does not depend on a certain coordinate, this coordinate is called cyclic (ignorable)

• For cyclic coordinates, conjugate momenta are conserved

)( iqfL

mqL

mq

0

ii qL

dtd

qL

0

iqL

dtd

Symmetries and conservation laws

• For cyclic coordinates, conjugate momenta are conserved

p =

cons

t p ≠ const


• Cyclic coordinates:

• Rotational symmetry Translational symmetry

• Conjugate momenta:

)(2

)( 2222

rVzrrmL

0

L

dtdL

constmr 2

0

zL

dtd

zL

constzm

z

Electromagnetism

• Conjugate momenta:

)(2

)( 222

Arqqrrrm

L zyx

jrL

jrm jqA jrm

Noether’s theorem

• Relationship between Lagrangian symmetries and conserved quantities was formalized only in 1915 by Emmy Noether:

• “For each symmetry of the Lagrangian, there is a conserved quantity”

• Let the Lagrangian be invariant under the change of coordinates:

• α is a small parameter. This invariance has to hold to the first order in α

),,...,,(~21 tqqqqq Niii

Emmy Noether/Amalie Nöther(1882 – 1935)

Noether’s theorem

• Invariance of the Lagrangian:

• Using the Euler-Lagrange equations

0ddL

N

i

i

i

i

i

qqLq

qL

ddL

1

~~

~~

N

ii

ii

i qL

qL

1~~

N

ii

ii

i qL

qL

dtd

1~~

N

ii

iqL

dtd

1~ 0

constqLN

ii

i

1

),,...,,(~21 tqqqqq Niii

Example

• Motion in an x-y plane of a mass on a spring (zero equilibrium length):

• The Lagrangian is invariant (to the first order in α) under the following change of coordinates:

• Then, from Noether’s theorem it follows that

2)(

2)( 2222

yxyx rrkrrmL

xyyyxx rrrrrr ~ ;~

constrL

rL

yy

xx

yxrrm xyrrm const

Example

• In polar coordinates:

• The conserved quantity:

• Angular momentum in the x-y plane is conserved

constrrmrrm xyyx

sincosrrrr

y

x

cossin

sincos

rrr

rrr

y

x

xyyx rrmrrm sin)sincos( rrrm

cos)cossin( rrrm 2mr const

Example

• For the same problem, we can start with a Lagrangian expressed in polar coordinates:

• The Lagrangian is invariant (to any order in α) under the following change of coordinates:

• The conserved quantity from Noether’s theorem:

constmr 12

2)(

2)( 2222

yxyx rrkrrmL

22)( 2222 krrrm

1~

constL


• How to find the special values for action corresponding to observable trajectories ?

• We look for a stationary action using variational principle

2

1

,)()]([t

ti

mi

dttdt

trdLtrI

)()(),( 0 ttrtr mmm

0)()( 21 tt mm 0)(

0

ddI

2

1

,),()],([)(t

ti

mi

m dttdttrdLtrII

),,...,,(~21 tqqqqq Nmmm

Stationary value

2

1

),...,,,()( x

x

dxxyyyfdd

ddJ

2

1

),...,,,(x

x

dxd

xyyydf

2

1

...x

x

dxyyfy

yfy

yf

1

2

3

2

1

.2x

x

dxyyf

2

1

2x

x

dxxy

yf

u

dv

2

1

x

x

yyf

u

v

2

1

x

x

dxyf

dxdy

v

du

)()(),( 0 xxyxy

2

1

x

xyf

2

1

x

x

dxyf

dxd

0)(0)(

2

1

xx

constqLN

ii

i

1

More on symmetries

• Full time derivative of a Lagrangian:

• From the Euler-Lagrange equations:

• If

dtdL

M

m

M

mm

mm

m

qqLq

qL

tL

1 1

M

m

M

mm

mm

m

qqLq

qL

dtd

tL

1 1

M

mm

m

qqL

dtd

tL

1

LqqL

dtd

tL M

mm

m1

dt

dH

0tL constLq

qLH

M

mm

m

1

What is H?

• Let us expand the Lagrangian in powers of :

• From calculus, for a homogeneous function f of degree n (Euler’s theorem) :

......),,...,,(

),,...,,(),,...,,(

3210,

212

211210

LLLLqqtqqql

qtqqqltqqqLL

jji

iMij

iiMiM

fnxfx

i ii

iq

...210

ii

iii

iii

iii

i

qqLq

qLq

qLq

qL

...320 321 LLL

What is H?

• If the Lagrangian has a form:

• Then

• For electromagnetism:

LqqLH

M

mm

m

1

...32 321 LLL

...)( 3210 LLLL ...2 320 LLL

210 LLLL

02 LLH

)(2/2 ArqqrmL

2L 0L 1L

02 LLH qrm 2/2 EVT

Conservation of energy

• In the field formalism, the conservation of H is a part of Noether’s theorem

210 LLLL

EH

constEtL

0


• Similarly to the “H-trick”:

2

112 dxft

gyy

f2

1 2 0

xf

0

1222

123

2

ygy

ydxd

gy

y

Scary!

constfyfy

H

gyy

ygy

yy2

1

12

2

2

constygy

212

1

21/ yyC

!!!


• Change of variables:

• Parametric solution (cycloid)

21/ yyC

2sinCy

dxdy

yCy 1 dy

yCydx

)sin(sin

sin 22

2

CdCC

Cdx

dC sin2 2

BdCx sin2 2

2sin

)2/)2(sin(

Cy

CBx

)2/)2(sin( CB

Scale invariance

• For Lagrangians of the following form:

• And homogeneous L0 of degree k

• Introducing scale and time transformations

• Then

jji

iijM qqlqqqLLLL ,

221020 ),...,,( constl ij 2

ttqq ii

''

),...,,(),...,,(' 2102100 Mk

M qqqLqqqLL

jji

iij qql

,2

2

ii qq

' j

jiiij qqlL '''

,22

Scale invariance

• Therefore, after transformations

• If

• Then

• The Euler-Lagrange equations after transformations

• The same!

2

2

0' LLL k

k

2

LL k'

0''

jj qL

dtd

qL

0)()(

j

k

j

k

qL

dtd

qL

0

jj qL

dtd

qL

Scale invariance

• So, the Euler-Lagrange equations after transformations are the same if

• Free fall

• Let us recall

k

2

2/1 k2/1

2/1 ''k

i

ik

qq

tt

mgzzmL 2

21k

2/1'' k

zz

tt

2/1'

zz

2/12 zgzt

Scale invariance


• Mass on a spring

• Let us recall

k

2

2/1 k2/1

2/1 ''k

i

ik

qq

tt

22

22 KzzmL

2k2/1'' k

zz

tt

0'

zz

02 zKmT

Scale invariance


• Kepler’s problem

• Let us recall 3rd Kepler’s law

k

2

2/1 k2/1

2/1 ''k

i

ik

qq

tt

2

2

2 r

MmGrmL 1k

2/1'' k

zz

tt

2/3'

zz

2/3RT

Johannes Kepler(1571-1630)

How about open systems?

• For some systems we can neglect their interaction with the outside world and formulate their behavior in terms of Lagrangian formalism

• For some systems we can not do it

• Approach: to describe the system without “leaks” and “feeds” and then add them to the description of the system

How about open systems?

• For open systems, we first describe the system without “leaks” and “feeds”

• After that we add “leaks” and “feeds” to the description of the system

• Q: Non-conservative generalized forces

jj qL

qL

dtd

jQ

Generalized forces

• Forces

• 1: Conservative (Potential)

• 2: Non-conservative

j

jj

Qq

Tq

Tdtd

UU

jjjjj

Qqqdt

dqT

qT

dtd

UU

...),,...,,( 21 tqqqV NU

12

1

UTL

Generalized forces

• In principle, there is no need to introduce generalized forces for a closed system fully described by a Lagrangian

• Feynman: “…The principle of least actiononly works for conservative systems — where all forces can be gotten from a potential function. … On a microscopic level — on the deepest level of physics — there are no non-conservative forces. Non-conservative forces, like friction, appear only because we neglect microscopic complications — there are just too many particles to analyze.”

• So, introduction of non-conservative forces is a result of the open-system approach


(1918 – 1988)

Degrees of freedom

• The number of degrees of freedom is the number of independent coordinates that must be specified in order to define uniquely the state of the system

• For a system of N free particle there are 3N degrees of freedom (3N coordinates)

N) ..., ,2 ,1(

ˆˆˆ

Ni

rkrjrir ziyixii

Constraints

• We can impose k constraints on the system

• The number of degrees of freedom is reduced to 3N – k = s

• It is convenient to think of the remaining s independent coordinates as the coordinates of a single point in an s-dimensional space: configuration space N

), ..., , ,(...

), ..., , ,(

321

32111

tqqqrr

tqqqrr

kNNN

kN

k

Types of constraints

• Holonomic (integrable) constraints can be expressed in the form:

• Nonholonomic constraints cannot be expressed in this form

• Rheonomous constraints – contain time dependence explicitly

• Scleronomous constraints – do not contain time dependence explicitly

kj

tqqqf nj

,...,2,1

0), ..., , ,( 21

Analysis of systems with holonomic constraints

• Elimination of variables using constraints equations

• Use of independent generalized coordinates

• Lagrange’s multiplier method

Double 2D pendulum

• An example of a holonomic scleronomous constraint

• The trajectories of the system are very complex

• Lagrangian approach produces equations of motion

• We need 2 independent generalized coordinates (N = 2, k = 2 + 2, s = 3 N – k = 2)

0)( 21

21 lr 0)( 2

22

21 lrr

1 2

Double 2D pendulum

• Relative to the pivot, the Cartesian coordinates

• Taking the time derivative, and then squaring

• Lagrangian in Cartesian coordinates:

11,1 sinlr x

11,1 coslr z 2211,2 sinsin llr x

2211,2 coscos llr z

21

21

21 lr

)sinsincos(cos2 212121212

22

22

12

12

2 llllr

)(2 ,22,11

222

211

zz rmrmgrmrmL

Double 2D pendulum

• Lagrangian in new coordinates:

• The equations of motion:

)coscos(cos

2)cos(2

222112111

2121212

22

22

12

122

12

11

llgmglm

llllmlmL

222212

1212

212

12122

22

22

1121212

2212

212

22122

12

121

sin)sin(

)cos(0

sin)()sin(

)cos()(0

glmllm

llmlm

glmmllm

llmlmm

Double 2D pendulum

• Special case

• The equations of motion:

• More fun at:

http://www.mathstat.dal.ca/~selinger/lagrange/doublependulum.html

21 mm 0, 21 lll 21

221

121

0

220

gl

gl

http://www.mathstat.dal.ca/~selinger/lagrange/doublependulum.html

Lagrange’s multiplier method

• Used when constraint reactions are the object of interest

• Instead of considering 3N - k variables and equations, this method deals with 3N + k variables

• As a results, we obtain 3N trajectories and k constraint reactions

• Lagrange’s multiplier method can be applied to some nonholonomic constraints


• Let us explicitly incorporate constraints into the structure of our system

• For observable trajectories

• So

kjtqqqf nj ,...,2,1 ;0), ..., , ,( 21

k

jnjj tqqqftLL

121 ), ..., , ,()('

0), ..., , ,( 21 tqqqf nj

k

jjj fLL

1

' L

ii qL

qL

dtd

''

k

j i

jj

ii qf

qL

qL

dtd

1

0

0


• - constraint reactions

• Now we have 3N + k equations for and

kjtqqqf nj ,...,2,1 ;0), ..., , ,( 21

k

j i

jj

ii qf

qL

qL

dtd

1

iQ

iQ

Niqf

qL

qL

dtd k

j i

jj

ii

3,...,2,1 ;1

iq j

Application to a nonholonomic case

• A particle on a smooth hemisphere

• One nonholonomic constraint:

• While the particle remains on the sphere, the constraint is holonomic

• And the reaction from the surface is not zero

02222 arrr zyx

02222 arrr zyx


• Constraint equation in cylindrical coordinates:

• New Lagrangian in cylindrical coordinates:

• Equations of motion

0 ar

)(cos2

)(' 1

2222

armgrzrrmL

rf

rL

dtd

rL

1

1

01cos 12 mgmrrm





0 ar

)(cos2

)(' 1

2222

armgrzrrmL

0

L

dtdL

0sin2 mgrmr





• Trivial

0 ar

)(cos2

)(' 1

2222

armgrzrrmL

0

zL

dtd

zL

0zm


• Constraint reaction:

ar

0cos 12 mgmrrm

0sin2 mgrmr

amg /cos 12

sinag

sin22ag

cos22

ag

dtd

dtd

0

0

)cos1(22

ag

)2cos3(1 mg

)2cos3(111

1 mgrf


• Constraint reaction:

• Reaction disappears when

• The particle becomes airborne

)2cos3(1 mg

2cos3

32cos 1

ar

Variational Principles and Lagrange’s Equations

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