Homework Help: Canonical transformation

1. Oct 16, 2012

aaaa202

There's a part in my book that I don't understand. I have attached the part and it is basically about how to transform from a set of conjugate variables (q,p) to another (Q,P) while preserving the hamilton equations of motion. I dont understand what he means by q,Q being separately independent. Don't we seek transformation where Q is a function of q. Maybe I'm just not into what he means by this independency.

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2. Oct 16, 2012

dextercioby

It's true, but he means that for F q and Q are variables, so that there's no linear dependence between their time derivatives. Such a linear dependence would spoil his argument.

3. Oct 16, 2012

aaaa202

Please elaborate. If Q = Q(q,p,t) how is its time derivative independent of q? :(

4. Oct 17, 2012

dextercioby

Well, what is said was that there is no linear dependence between $\displaystyle{\dot{Q}}$ and $\displaystyle{\dot{q}}$.

5. Oct 17, 2012

aaaa202

but dQ/dt = $\partial$Q/$\partial$q dq/dt + .....
How is that not a relation between them?