canonical pb

8/13/2019 Canonical Pb

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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


7/25

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


9/25

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

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


10/25

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


11/25

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


22/25

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


23/25

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


p p H H H p p H

q p p q q


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

canonical pb

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