Register to reply

Canonical Transform proof

by Liquidxlax
Tags: canonical, proof, transform
Share this thread:
Mar3-12, 04:09 PM
P: 322
Consider the following change of variables in phase space f: maps the reals and is smooth and invertable change of coordinates Q=f(q), q = f-1(Q). Given f, define a change of variables on phase space (q,p) -> (Q,P) by the pair of relations

Pj = (∇f-1)Tjk(f(q))pk

q runs from 1 to n

show that its canonical.

I know that for this to be canonical

(dQi/dqj)(q,p) = (dpj/dPi)(Q,P)

(dQi/dpj)(q,p) = -(dqj/dPi)(Q,P)

i'm having a couple problems, is f the generating function that i have to find explicitly?

Can i use the sympletic method such that MJM^T = J

what is the point of the transpose for the (∇f-1)Tjk part for? i thought f had to be symmetric.
Phys.Org News Partner Science news on
Scientists develop 'electronic nose' for rapid detection of C. diff infection
Why plants in the office make us more productive
Tesla Motors dealing as states play factory poker
Mar4-12, 11:11 AM
P: 322
for Pj it can equal


=(∇f)T *-1jkqpk

but for that to be true then (∇f-1)Tjk has to be symmetric therefore the transpose dissapears

looking at my notes, i think this is supposed to be the associated point transformation in phase space
Mar5-12, 12:04 PM
P: 322
Guess i can answer my own question... i knew how to do it but i had an error in my notes

[dQ, dP)T = [{dQ/dq, dQ/dp}, {dP/dq, dP/dp}]*[dq, dp] = Mij*[dq, dp]

to prove its a canonical transformation

MJMT = J where J = [{0,I},{-I,0}] and T represents the transpose

If i do the matrix multiplication and say:

now i will let {a,b} where a and b are some arbitrary coordinates be the poisson brackets

MJMT = [(0,{Q,P}), (-{Q,P},0)]

It is known that {P,P} = 0 = {Q,Q} and {Q,P} = δij

Using the following vvvv

and there you have it

Register to reply

Related Discussions
Canonical Map Proof Calculus & Beyond Homework 1
Microcanonical vs canonical vs grand canonical ensemble Classical Physics 1
Laplace transform of the grand canonical partition function Atomic, Solid State, Comp. Physics 1
Jordan Canonical form proof Linear & Abstract Algebra 20
Is it true that unitary transform in QM corresponds to canonical transform Quantum Physics 1