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(adsbygoogle = window.adsbygoogle || []).push({}); ^{-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.

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# Canonical Transform proof

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