lecture21 canonical transformation
DESCRIPTION
Lecture on Analytical Mechanics. Notes by Masahiro Morii at Harvard.TRANSCRIPT
MechanicsPhysics 151
Lecture 21Canonical Transformations
(Chapter 9)
What We Did Last Time
Canonical transformationsHamiltonian formalism isinvariant under canonical + scale transformationsGenerating functions define canonical transformationsFour basic types of generating functions
They are all practically equivalent
Used it to simplify a harmonic oscillatorInvariance of phase space
i i i idFPQ K p q Hdt
− + = −
1( , , )F q Q t 2 ( , , )F q P t 3 ( , , )F p Q t 4 ( , , )F p P t
Four Basic Generators
Trivial CaseDerivativesGenerator
1( , , )F q Q t
2 ( , , ) i iF q P t Q P−
3 ( , , ) i iF p Q t q p+
4 ( , , ) i i i iF p P t q p Q P+ −
1i
i
Fpq
∂=
∂1
ii
FPQ
∂= −
∂ 1 i iF q Q= i iQ p=
i iP q= −
2i
i
Fpq
∂=
∂2
ii
FQP
∂=
∂ 2 i iF q P=i iP p=i iQ q=
3i
i
Fqp
∂= −
∂3
ii
FPQ
∂= −
∂
4i
i
Fqp
∂= −
∂4
ii
FQP
∂=
∂
3 i iF p Q=i iP p= −i iQ q= −
4 i iF p P= i iQ p=
i iP q= −
Goals for Today
Dig deeper into Canonical TransformationsInfinitesimal Canonical Transformation
Very small changes in q and pDefine generator G for an ICT
Direct Conditions for Canonical TransformationNecessary-and-sufficient conditions for any CT
Poisson BracketInvariant of any Canonical TransformationConnect to Infinitesimal Canonical Transformation
Infinitesimal CT
Consider a CT in which q, p are changed by small (infinitesimal) amounts
ICT is close to identity transf.Generating function should be
i i iQ q qδ= + i i iP p pδ= + Infinitesimal Canonical Transformation (ICT)
2 ( , , ) ( , , )i iF q P t q P G q P tε= +
Identity CT generator Small
2i i
i i
F Gp Pq q
ε∂ ∂= = +
∂ ∂2
i ii i
F GQ qP P
ε∂ ∂= = +
∂ ∂Look at the
generator table
ii i
G GqP p
δ ε ε∂ ∂= ≈
∂ ∂ ii i
G Gpq Q
δ ε ε∂ ∂= − ≈ −
∂ ∂Since ε is
infinitesimal
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 terms of q or Q, p or P, interchangeably
For example:
2 ( , , ) ( , , )i iF q P t q P G q P tε= +
i ii
GQ qP
ε ∂= +
∂ i ii
GP pq
ε ∂= −
∂
( , , )G G q p t= i ii
GQ qp
ε ∂= +
∂ i ii
GP pq
ε ∂= −
∂
Hamiltonian
Consider
What does ε look like? Infinitesimal time δt
Hamiltonian is the generator of infinitesimal time transformation
In QM, you learn that Hamiltonian is the operator that represents advance of time
( , , )G H q p t=
i ii
Hq qp
δ ε ε∂= =
∂ i ii
Hp pq
δ ε ε∂= − =
∂
i iq q tδ δ= i ip p tδ δ=
Direct Conditions
Consider a restricted Canonical TransformationGenerator has no t dependence
Q and P depends only on q and p
0Ft
∂=
∂( , ) ( , )K Q P H q p= Hamiltonian
is unchanged
( , )i iQ Q q p= ( , )i iP P q p=
i i i ii j j
j j j j j j
Q Q Q QH HQ q pq p q p p q
∂ ∂ ∂ ∂∂ ∂= + = −
∂ ∂ ∂ ∂ ∂ ∂
i i i ii j j
j j j j j j
P P P PH HP q pq p q p p q
∂ ∂ ∂ ∂∂ ∂= + = −
∂ ∂ ∂ ∂ ∂ ∂
Hamilton’s equations
Direct Conditions
On the other hand, Hamilton’s eqns say
i ii
j j j j
Q QH HQq p p q
∂ ∂∂ ∂= −
∂ ∂ ∂ ∂
i ii
j j j j
P PH HPq p p q
∂ ∂∂ ∂= −
∂ ∂ ∂ ∂
j ji
i j i j i
q pH H HQP q P p P
∂ ∂∂ ∂ ∂= = +
∂ ∂ ∂ ∂ ∂
j ji
i j i j i
q pH H HPQ q Q p Q
∂ ∂∂ ∂ ∂= − = − −
∂ ∂ ∂ ∂ ∂
Direct Conditions
for a Canonical Transformation
,,
ji
j i Q Pq p
pQq P
⎛ ⎞ ∂⎛ ⎞∂=⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠⎝ ⎠ ,,
ji
j i Q Pq p
qQp P
⎛ ⎞ ∂⎛ ⎞∂= −⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠⎝ ⎠
,,
ji
j i Q Pq p
pPq Q
⎛ ⎞ ∂⎛ ⎞∂= −⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠⎝ ⎠ ,,
ji
j i Q Pq p
qPp Q
⎛ ⎞ ∂⎛ ⎞∂=⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠⎝ ⎠
Direct Conditions
Direct Conditions are necessary and sufficient for a time-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 doesn’t work
,,
ji
j i Q Pq p
pQq P
⎛ ⎞ ∂⎛ ⎞∂=⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠⎝ ⎠ ,,
ji
j i Q Pq p
qQp P
⎛ ⎞ ∂⎛ ⎞∂= −⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠⎝ ⎠
,,
ji
j i Q Pq p
pPq Q
⎛ ⎞ ∂⎛ ⎞∂= −⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠⎝ ⎠ ,,
ji
j i Q Pq p
qPp Q
⎛ ⎞ ∂⎛ ⎞∂=⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠⎝ ⎠
Infinitesimal CT
Does an ICT satisfy the DCs?2( )i i i
ijj j i j
Q q q Gq q P q
δ δ ε∂ ∂ + ∂= = +
∂ ∂ ∂ ∂
2( )j j jij
i i i j
p P p GP P P q
δδ ε
∂ ∂ − ∂= = +
∂ ∂ ∂ ∂
ii i
G GqP p
δ ε ε∂ ∂= ≈
∂ ∂
ii i
G Gpq Q
δ ε ε∂ ∂= − ≈ −
∂ ∂
2( )i i i
j j i j
Q q q Gp p P p
δ ε∂ ∂ + ∂= =
∂ ∂ ∂ ∂
2( )j j j
i i i j
q Q q GP P P p
δε
∂ ∂ − ∂= = −
∂ ∂ ∂ ∂2( )i i i
j j i j
P p p Gq q Q q
δ ε∂ ∂ + ∂= = −
∂ ∂ ∂ ∂
2( )j j j
i i i j
p P p GQ Q Q q
δε
∂ ∂ − ∂= =
∂ ∂ ∂ ∂2( )i i i
ijj j i j
P p p Gp p Q p
δ δ ε∂ ∂ + ∂= = −
∂ ∂ ∂ ∂
2( )j j jij
i i i j
q Q q GQ Q Q p
δδ ε
∂ ∂ − ∂= = −
∂ ∂ ∂ ∂
Yes!
Successive CTs
Two successive CTs make a CT
Direct Conditions can also be “chained”, e.g.,
1i i i i
dFPQ K p q Hdt
− + = − 2i i i i
dFY X M PQ Kdt
− + = −
1 2( )i i i i
d F FY X M p q Kdt+
− + = − True for unrestricted CTs
,,
ji
j i Q Pq p
pQq P
⎛ ⎞ ∂⎛ ⎞∂=⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠⎝ ⎠ ,,
ji
j i X YQ P
PXQ Y
⎛ ⎞ ∂⎛ ⎞∂=⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠⎝ ⎠
,,
ji
j i X Yq p
pXq Y
⎛ ⎞ ∂⎛ ⎞∂=⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠⎝ ⎠
Easy to prove
Unrestricted CT
Now we consider a general, time-dependent CT
Let’s 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 iP P q p t=FK Ht
∂= +
∂
,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
Unrestricted CT
Concentrate on a time-only CTBreak t – t0 into pieces of infinitesimal time dt
Each step is an ICT Satisfies Direct Conditions“Integrating” gives us what we needed
The proof worked because a time-only CT is a continuous transformation, parameterized by t
( ), ( )Q t P t0 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 the Direct Conditions, and vice versa
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
Let’s start with a few basic rules
[ ] ,,
q pi i i i
u v u vu vq p p q
∂ ∂ ∂ ∂≡ −
∂ ∂ ∂ ∂Poisson Bracket
[ ]1 1, ( )u v uv vui i
≡ − for two operators u and v
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+ + =
Jacobi’s Identity
All easy to prove
This one is worth trying.See Goldstein if you are lost
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?
[ , ] 0j jk k
i ij k
i i
q qqp q
q q qq p
∂ ∂∂ ∂= −
∂ ∂ ∂=
∂[ , ] 0j kp p =
[ , ] j jk k
i i i ij k jk
q qp pq p q
q pp
δ∂ ∂∂ ∂
=∂ ∂
=−∂ ∂
[ , ]j k jkp q δ= −
, ,q p Q P→
Fundamental PB and CT
Fundamental Poisson Brackets are invariant under CT
,[ , ] 0j j j j jk k i ij k q p
i i i i i k i k k
Q Q Q Q QQ Q q pQ Qq p p q q P p P P
∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂= − = − − = − =
∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂
,[ , ] 0j j j j jk k i ij k q p
i i i i i k i k k
P P P P PP P q pP Pq p p q q Q p Q Q
∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂= − = + = =
∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂
,[ , ] j j j j jk k i ij k q p jk
i i i i i k i k k
Q Q Q Q QP P q pQ Pq p p q q Q p Q Q
δ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂
= − = + = =∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂
,[ , ] [ , ]j k q p k j jkP Q Q P δ= − = − Used Direct Conditions here
Poisson Bracket and CT
What happens to a Poisson Bracket under CT?For a time-independent CT
[ ] ,
, ,
,
[ , ] [ , ] [
j j j jk k k k
j i j i k i k i j i j i k i k i
j k j k j k
Q Pi i i i
j k Q P j k Q P
q p q pq p q pu u v v u u v v
q Q p Q q P p P q P p P q Q p Q
u v u v u v
q q q p p q
u v u vu vQ P P Q
q q q p
∂ ∂ ∂ ∂∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂+ + − + +
∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂
∂ ∂ ∂ ∂ ∂ ∂
∂ ∂ ∂ ∂ ∂ ∂
∂ ∂ ∂ ∂≡ −
∂ ∂ ∂ ∂
⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞= ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠
= + +
[ ]
, ,
,
, ] [ , ]
,
j k
j k j k
j k Q P j k Q P
jk jk
q p
u v
p p
u v u v
q p p q
p q p p
u v
δ δ
∂ ∂
∂ ∂
∂ ∂ ∂ ∂
∂ ∂ ∂ ∂
+
= −
= Poisson Brackets are invariant under CT
Invariance of Poisson Bracket
Poisson Brackets are canonical invariantsTrue for any Canonical Transformations
Goldstein shows this using “simplectic” approach
We don’t have to specify q, p in each PB[ ] ,
,q p
u v [ ],u v good enough
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 qp
ε ∂= +
∂ i ii
GP pq
ε ∂= −
∂
[ , ] i ii i
j j j j i
q qG G Gq G qq p p q p
ε ε ε δ⎛ ⎞∂ ∂∂ ∂ ∂
= − = =⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠
[ , ] i ii i
j j j j i
p pG G Gp G pq p p q q
ε ε ε δ⎛ ⎞∂ ∂∂ ∂ ∂
= − = − =⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠
ICT and Poisson Bracket
For an arbitrary function u(q,p,t), the ICT does
That is
[ , ]
ICTi i
i i
i i i i
u u uu u u u q p tq p tu G u G uu tq p p q t
uu u G tt
δ δ δ δ
ε ε δ
ε δ
∂ ∂ ∂⎯⎯⎯→ + = + + +
∂ ∂ ∂∂ ∂ ∂ ∂ ∂
= + − +∂ ∂ ∂ ∂ ∂
∂= + +
∂
[ , ] uu u G tt
δ ε δ∂= +
∂
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,
[ , ] uu t u H tt
δ δ δ∂= +
∂[ , ]du uu H
dt t∂
= +∂
[ , ] 0uu Ht
∂+ =
∂
[ , ] uH ut
∂=
∂u is a constant of motion
Infinitesimal Time Transf.
If u does not depend explicitly on time,
Try this on q and p
[ , ] [ , ]du uu H u Hdt t
∂= + =
∂
[ , ] i ii i
j j j j i
p pH H Hp p Hq p p q q
∂ ∂∂ ∂ ∂= = − = −
∂ ∂ ∂ ∂ ∂
[ , ] i ii i
j j j j i
q qH H Hq q Hq p p q p
∂ ∂∂ ∂ ∂= = − =
∂ ∂ ∂ ∂ ∂ Hamilton’sequations!
Summary
Direct ConditionsNecessary and sufficientfor Canonical Transf.
Infinitesimal CTPoisson Bracket
Canonical invariantFundamental PB
ICT expressed by
Infinitesimal time transf. generated by Hamiltonian
,,
ji
j i Q Pq p
pQq P
⎛ ⎞ ∂⎛ ⎞∂=⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠⎝ ⎠ ,,
ji
j i Q Pq p
qQp P
⎛ ⎞ ∂⎛ ⎞∂= −⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠⎝ ⎠
,,
ji
j i Q Pq p
pPq Q
⎛ ⎞ ∂⎛ ⎞∂= −⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠⎝ ⎠ ,,
ji
j i Q Pq p
qPp Q
⎛ ⎞ ∂⎛ ⎞∂=⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠⎝ ⎠
[ ],i i i i
u v u vu vq p p q
∂ ∂ ∂ ∂≡ −
∂ ∂ ∂ ∂
[ , ] [ , ] 0i j i jq q p p= = [ , ] [ , ]i j i j ijq p p q δ= − =
[ , ] uu u G tt
δ ε δ∂= +
∂