Are All Canonical Transformations Governed by the Generating Function Relation?

[A_B]

Hi,

Im working through some chapters of Goldstein and I'm up to canonical transformations now. On page 370 it says that the variational principle for the hamiltonians K and H are both satisfied if H and K are connected by a relation of the form

λ(pq' - H) = PQ' - K + dF/dt

And I can see this. My question is, are there canonical transformations that do not fit this relation? And if so are they impportant? Is this relation a very general one, or does it simply turn out to give good tranformations in many problems?

A_B

 

[A_B]

Also if we obtain a relation

pq' - H = PQ' - K + ∂F/∂t + (∂F/∂q)q' + (∂F/∂Q)Q'

Goldstein says (p 372 eq 9.13)

"Since the old and the new coordinates, q and Q, ere separately independent, the above equation can hold identically only if the coefficients of q' and Q' each vanish:"

leading to

p = ∂F/∂q,

P = -∂F/∂Q

I don't see why this is necessary for the relation to hold.

 

[Matterwave]

Hi,

Im working through some chapters of Goldstein and I'm up to canonical transformations now. On page 370 it says that the variational principle for the hamiltonians K and H are both satisfied if H and K are connected by a relation of the form

λ(pq' - H) = PQ' - K + dF/dt

And I can see this. My question is, are there canonical transformations that do not fit this relation? And if so are they impportant? Is this relation a very general one, or does it simply turn out to give good tranformations in many problems?

A_B

Actually the definition of canonical transformation usually stipulates that λ=1 in the above requirement. For any other λ, we call these "extended canonical transformations".

This is very general because otherwise the action would not remain stationary for the actual motion. This is because the freedom in the Lagrangian is only in the total time derivative term.

 

[A_B]

can someone answer my question in the second post?

thanks!
A_B