PHY 711 Fall 2013 -- Lecture 13 19/25/2013

PHY 711 Classical Mechanics and Mathematical Methods

10-10:50 AM MWF Olin 103

Plan for Lecture 13:Finish reading Chapter 6

1. Canonical transformations2. Hamilton-Jacobi formalism

PHY 711 Fall 2013 -- Lecture 13 29/25/2013

PHY 711 Fall 2013 -- Lecture 13 39/25/2013

PHY 711 Fall 2013 -- Lecture 13 49/25/2013

PHY 711 Fall 2013 -- Lecture 13 59/25/2013

Review:Liouville’s theorem: Imagine a collection of particles obeying the Canonical equations of motion in phase space.

in timeconstant is 0

: thatshows theormsLiouville',,

:space phasein particles of on"distributi" thedenote Let

3131

DdtdD

tppqqDDD

NN

PHY 711 Fall 2013 -- Lecture 13 69/25/2013

Proof of Liouville’e theorem:

tD

v

v

DtD v

:equation Continuity

N21N21

,,,,,:gradient ldimensiona 6 a have also We

,,,,,: vectorldimensiona 6 theis velocity thecase, in this :Note

2121

ppprrr

ppprrrv

N

N

NN

PHY 711 Fall 2013 -- Lecture 13 79/25/2013

N

j j

j

j

jN

jj

jj

j

N

jj

jj

j

pp

qq

DppDq

qD

Dpp

Dqq

DtD

3

1

3

1

3

1

v

022

jjjjj

j

j

j

qpH

pqH

pp

qq

PHY 711 Fall 2013 -- Lecture 13 89/25/2013

03

1

3

1

dtdDp

pDq

qD

tD

ppDq

qD

tD

N

jj

jj

j

N

jj

jj

j

N

j j

j

j

jN

jj

jj

j pp

qq

DppDq

qD

tD 3

1

3

1

0

PHY 711 Fall 2013 -- Lecture 13 99/25/2013

Notion of “Canonical” distributions

tQqFdtdtPQHQP

tpqHqp

tPPQQpptPPQQqq

nn

nn

,,,,~

,,

each for ,,each for ,,

11

11

~

~

0~

:principle sHamilton'Apply

σσ

σσ

t

tσσσσ

σσσ

QHP

PHQ

dt,tQ,qFdtd,tP,QHQP

f

i

PHY 711 Fall 2013 -- Lecture 13 109/25/2013

Note that it is conceivable that if we were extraordinarily clever, we could find all of the constants of the motion!

tFQ

QFq

qFtQqF

dtd

tQqFdtdtPQHQP

tpqHqp

,,

,,,,~

,,

tFtPQHQ

QFP

tpqHqqFp

,,~

,,

PHY 711 Fall 2013 -- Lecture 13 119/25/2013

tFtpqHtPQH

QFP

qFp

tFtPQHQ

QFP

tpqHqqFp

,,,,~

,,~

,,

PHY 711 Fall 2013 -- Lecture 13 129/25/2013

Note that it is conceivable that if we were extraordinarily clever, we could find all of the constants of the motion!

motion theof constants are ,

0~

and 0~

:Suppose

~

~

σσ

σσ

σσ

σσ

σσ

PQQHP

PHQ

QHP

PHQ

Possible solution – Hamilton-Jacobi theory:

tPqSQPtQqF ,,,, :Suppose

PHY 711 Fall 2013 -- Lecture 13 139/25/2013

tStpqHtPQH

PSQ

qSp

tSP

PSq

qSQPtPQH

tPqSQPdtdtPQHQP

tpqHqp

,,,,~

:Solution

,,~

,,,,~

,,

PHY 711 Fall 2013 -- Lecture 13 149/25/2013

0,,

,, find toNeed 0~ choose constants; are ~,, Assume

:clearsdust When the

tSt

qSqH

PSQ

qSp

tPqSHHPQ

tPqSQPdtdtPQHQP

tpqHqpS

,,,,~

,,:action"" theis :Note

00 0

PHY 711 Fall 2013 -- Lecture 13 159/25/2013

tPqSQPdtdtPQHQP

tpqHqp

,,,,~

,,

00 0

f

i

f

i

f

i

t

t

t

t

t

t

tPqS

dttPqSdtddttpqHqp

,,

,,,,

PHY 711 Fall 2013 -- Lecture 13 169/25/2013

Differential equation for S:

0,,

tSt

qSqH

constant) ( )(),( :Assume

021

21

0,, :Eq Jacobi-Hamilton

21

2,, :Example

222

222

EEtqWtqStSqm

qS

m

tSt

qSqH

qmm

ptpqH

PHY 711 Fall 2013 -- Lecture 13 179/25/2013

dqqmmEqW

qmmEdqdW

EqmdqdW

m

EEtqWtqStSqm

qS

m

22

22

222

222

2)(

2

21

21

constant) ( )(),( :Assume

021

21

Continued:

PHY 711 Fall 2013 -- Lecture 13 189/25/2013

Qtm

mEtq

tmE

qmQES

EtmE

qmEqmmEqtEqS

CmE

qmEqmmEq

dqqmmEqW

sin2)(

2sin1

2sin2

21),,(

2sin2

21

2)(

1

122

122

22

Continued: