phy 7 11 classical mechanics and mathematical methods 10-10:50 am mwf olin 103
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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
DppDq
qD
Dpp
Dqq
DtD
3
1
3
1
3
1
v
022
jjjjj
j
j
j
qpH
pqH
pp
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
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:
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