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What We Did Last Time
Canonical transformationsHamiltonian formalism isinvariant under canonical + scale transformations
Generating functions define canonical transformationsFour basic types of generating functions
They are all practically equivalent
Used it to simplify a harmonic oscillator Invariance of phase space
i i i i
dF PQ K p q H
dt
1( , , ) F q Q t 2 ( , , ) F q P t 3 ( , , ) F p Q t 4 ( , , ) F p P t
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Four Basic Generators
Trivial CaseDerivativesGenerator
1( , , ) F q Q t
2 ( , , ) i i F q P t Q P
3 ( , , ) i i F p Q t q p
4 ( , , ) i i i i F p P t q p Q P
1i
i
F p
q1
i
i
F P
Q
1 i i F q Q i iQ p
i i P q
2i
i
F p
q2
ii
F Q
P 2 i i F q P
i i P pi iQ q
3i
i
F q
p
3i
i
F P
Q
4i
i
F q
p 4
ii
F Q
P
3 i i F p Qi i P p
i iQ q
4 i i F p P i iQ p
i i P q
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Generator of ICT
An ICT is generated by
G is called (inaccurately) the generator of the ICTSince the CT is infinitesimal, G may be expressed in termsof q or Q, p or P , interchangeably
For example:
2 ( , , ) ( , , )i i F q P t q P G q P t
i ii
GQ q
i i
i
G P p
q
( , , )G G q p t i i i
GQ q
p
i i i
G P p
q
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Hamiltonian
Consider
What does look like? Infinitesimal time t
Hamiltonian is the generator of infinitesimal time
transformationIn QM, you learn that Hamiltonian is the operator thatrepresents advance of time
( , , )G H q p t
i ii
H q q
p
i ii
H p p
q
i iq q t i i p p t
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Direct Conditions
Consider a restricted Canonical TransformationGenerator has no t dependence
Q and P depends only on q and p
0 F
t ( , ) ( , ) K Q P H q p Hamiltonian
is unchanged
( , )i iQ Q q p ( , )i i P q p
i i i i
i j j j j j j j
Q Q Q Q H H Q q pq p q p p q
i i i ii j j
j j j j j j
P P P P H P q p
q p q p p q
Hamiltonsequations
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Direct Conditions
On the other hand, Hamiltons eqns say
i ii
j j j j
Q Q H H Q
q p p q
i ii
j j j
P P H H P q p p q
ji
i j i j i
q p H H H Q
q P p P
j ji
i j i j i
q p H H H P Q q Q p Q
DirectConditions
for a CanonicalTransformation
,,
ji
j i Q P q p
pQ
q P
,,
ji
j i Q P q p
p P
,,
ji
j i Q P q p
p P q Q
,,
ji
j i Q P q p
q P p Q
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Direct Conditions
Direct Conditions are necessary and sufficient for atime-independent transformation to be canonical
You can use them to test a CT
In fact, this applies to all Canonical TransformationsBut the proof on the last slide doesnt work
,,
ji
j i Q P q p
pQq P
,,
ji
j i Q P q p
qQ p P
,,
ji
j i Q P q p
p P q Q
,,
ji
j i Q P q p
q P p Q
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Infinitesimal CT
Does an ICT satisfy the DCs?2( )i i i
ij j j i j
Q q q Gq q P q
2( ) j j jij
i i i j
p P p G P P P q
ii i
G Gq
P p
ii i
G G pq Q
2( )i i i j i j
Q q q G p p P p
2( ) j j j
i i i j
q Q q G P P P p
2( )i i i
j j i j
P p p Gq q Q q
2( ) j j j
i i i j
p P p GQ Q Q q
2( )i i iij
j j i j
P p p G p p Q p
2( ) j j jij
i i i j
q Q q GQ Q Q p
Yes!
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Successive CTs
Two successive CTs make a CT
Direct Conditions can also be chained, e.g.,
1i i i i
dF PQ K p q H
dt 2
i i i i
dF Y X M PQ K
dt
1 2( )i i i i
d F F Y X M p q K dt
True for unrestricted CTs
,,
ji
j i Q P q p
pQq P
,,
ji
j i X Y Q P
P X Q Y
,,
ji
j i X Y q p
p X q Y
Easy to prove
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Unrestricted CT
Now we consider a general, time-dependent CT
Lets do it in two steps
First step is t -independent Satisfies the DCsWe must show that the second step satisfies the DCs
( , , )i iQ Q q p t ( , , )i i P q p t F K H
t
,q p 0 0( , , ), ( , , )Q q p t P q p t ( , , ), ( , , )Q q p t P q p t
Time-independent CT Time-only CT
Fixed time
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Unrestricted CT
Concentrate on a time-only CTBreak t t 0 into pieces of infinitesimal time dt
Each step is an ICT Satisfies Direct ConditionsIntegrating gives us what we needed
The proof worked because a time-only CT is a continuoustransformation, parameterized by t
( ), ( )Q t P t 0 0( ), ( )Q t P t
0 0( ), ( )Q t P t 0 0( ), ( )Q t dt P t dt ( ), ( )Q t P t
All Canonical Transformations satisfies theDirect Conditions, and vice versa
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Poisson Bracket
For u and v expressed in terms of q and p
This weird construction has many useful featuresIf you know QM, this is analogous to the commutator
Lets start with a few basic rules
,, q pi i i i
u v u vu v
q p p q
Poisson Bracket
1 1, ( )u v uv vui i
for two operators u and v
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Poisson Bracket Identities
For quantities u, v, w andconstants a , b[ , ] 0u u
,, q pi i i i
u v u vu vq p p q
[ , ] [ , ]u v v u
[ , ] [ , ] [ , ]au bv w a u w b v w
[ , ] [ , ] [ , ]uv w u w v u v w
[ ,[ , ]] [ ,[ , ]] [ ,[ , ]] 0u v w v w u w u v
Jacobis Identity
All easy to prove
This one is worth trying.See Goldstein if you are lost
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Fundamental Poisson Brackets
Consider PBs of q and p themselves
Called the Fundamental Poisson Brackets
Now we consider a Canonical Transformation
What happens to the Fundamental PB?
[ , ] 0 j jk k i i
j k i i
q qq p q
q q q
q p
[ , ] 0 j k p p
[ , ] j jk k i i i i
j k jk
q q p pq p q
q p p
[ , ] j k jk p q
, ,q p Q P
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Fundamental PB and CT
Fundamental Poisson Brackets are invariant under CT
,[ , ] 0 j j j j jk k i i
j k q pi i i i i k i k k
Q Q Q Q QQ Q q pQ Q
q p p q q P p P P
,[ , ] 0 j j j j jk k i i
j k q pi i i i i k i k k
P P P P P P P q p P P
q p p q q Q p Q Q
,[ , ] j j j j jk k i i
k q p jki i i i i k i k k
Q Q Q Q Q P P q pQ P
q p p q q Q p Q Q
,[ , ] [ , ]
j k q p k j jk P Q Q P Used Direct Conditions here
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Invariance of Poisson Bracket
Poisson Brackets are canonical invariantsTrue for any Canonical Transformations
Goldstein shows this using simplectic approach
We dont have to specify q, p in each PB ,, q pu v ,u v good enough
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ICT and Poisson Bracket
Infinitesimal CT can be expressed neatly with a PB
For a generator G,
On the other hand
We can generalize further
i ii
GQ q
p
i ii
G P p
q
[ , ] i ii i j j j j i
q qG G Gq G q
q p p q p
[ , ] i i
i i j j j j i
p pG G G p G pq p p q q
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ICT and Poisson Bracket
For an arbitrary function u(q, p,t ), the ICT does
That is
[ , ]
ICT i i
i i
i i i i
u u uu u u u q p t
q p t
u G u G uu t q p p q t
uu u G t
t
[ , ] uu u G t t
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Infinitesimal Time Transf.
Hamiltonian generates infinitesimal time transf.Applying the Poisson Bracket rule
Have you seen this in QM?
If u is a constant of motion,
That is,
[ , ] u
u t u H t t
[ , ]du uu H dt t
[ , ] 0u
u H t
[ , ] u H ut
u is a constant of motion
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Infinitesimal Time Transf.
If u does not depend explicitly on time,
Try this on q and p
[ , ] [ , ]du u
u H u H dt t
[ , ] i ii i j j j j i
p p H H H p p H
q p p q q
[ , ] i ii i j j j j i
q q H H H q q H
q p p q p
Hamiltons
equations!
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Summary
Direct Conditions Necessary and sufficientfor Canonical Transf.
Infinitesimal CTPoisson Bracket
Canonical invariantFundamental PB
ICT expressed by
Infinitesimal time transf. generated by Hamiltonian
,,
ji
j i Q P q p
pQq P
,,
ji
j i Q P q p
qQ p P
,,
ji
j i Q P q p
p P q Q
,,
ji
j i Q P q p
q P p Q
,i i i i
u v u vu v
q p p q
[ , ] [ , ] 0i j i jq q p p [ , ] [ , ]i j i j ijq p p q
[ , ] u
u u G t t