Direct Approach
The functional form of Hamilton's equations is
By definition, the transformed coordinates have analogous dynamics
where K(Q,P) is a new Hamiltonian that must be determined.
In general, a transformation (q,p,t) → (Q,P,t) does not preserve the form of Hamilton's equations. For time independent transformations between (q,p) and (Q,P) we may check if the transformation is restricted canonical, as follows. Since restricted transformations have no explicit time dependence (by construction), the time derivative of a new generalized coordinate Qm is
where is the Poisson bracket.
We also have the identity for the conjugate momentum Pm
If the transformation is canonical, these two must be equal, resulting in the equations
The analogous argument for the generalized momenta Pm leads to two other sets of equations
These are the direct conditions to check whether a given transformation is canonical.
Read more about this topic: Canonical Transformation
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