# Canonical transformation

Lotophage
Problem: Verify that the infinitesimal transformation generated by any dynamical variable g is a canonical transformation.

I've worked out that an infinitesimal canonical transformation can be represented as follows:

q_i -> q_i' = q_i + &epsilon;(&part;g/&part;p_i) &equiv;q_i + &delta;q_i

p_i -> p_i' = p_i - &epsilon;(&part;g/&part;q_i) &equiv;p_i + &delta;p_i

I also know that, for any Hamiltonian, a new set of coordinates is only canonical if

{q_j',q_k'} = 0 = {p_j',p_k'}

==> {q_j',p_k'} = &delta;_jk

where {} are Poisson brackets and the lower scores denote subscript for the following character.

So my question is: Can I use these relationships to solve the problem? If so, how do I get started? The use of indices in these relationships confuses me, and I don't know if the &delta;_jk is supposed to be Kronecker delta, or some infinitesimal number. My intuition says that it's the Kronecker, but the indices used in the book I'm using are not explained, and I am unsure of how to apply the criterion for a canonical transformation to the general form of an infinitesimal transformation as given above. Any hints would be greatly appreciated!

Thanks,

Chris