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Is canonical transformation in mechanics unique? We know that given [tex]\ (q, p)\rightarrow\ (Q, P)[/tex], [tex]\ [q,p] = [Q,P] = constant[/tex] and Hamilton's equations of motion stay the same in the new co-ordinates.

My question is: given [tex]\ q\rightarrow Q[/tex] in a canonical transoformation, is the map [tex]\ p\rightarrow P[/tex] uniquely determined? Seems yes to me, but I do not find an off-hand argument in favour.

Can anyone tell how to derive this map?

-Regards,

Neel