Canonical Transform proof

Liquidxlax

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

P_j = (∇f^-1)^T_jk(f(q))p_k

q runs from 1 to n

show that its canonical.



I know that for this to be canonical

(dQ_i/dq_j)_(q,p) = (dp_j/dP_i)_(Q,P)

(dQ_i/dp_j)_(q,p) = -(dq_j/dP_i)_(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)^T_jk part for? i thought f had to be symmetric.

Liquidxlax

for P_j it can equal

(∇f^-1)^T_jkQp_k

=(∇f)^{T *-1}_jkqp_k

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

Liquidxlax

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] = M_ij*[dq, dp]

to prove its a canonical transformation

MJM^T = 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

MJM^T = [(0,{Q,P}), (-{Q,P},0)]

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

Using the following vvvv


and there you have it