Understanding the Concept of Canonical Transformation in Hamiltonian Mechanics

[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 don't 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.

 

[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.

 

[aaaa202]

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

 

[dextercioby]

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

 

[aaaa202]

but dQ/dt = \partialQ/\partialq dq/dt + ...
How is that not a relation between them?