Canonical Evolution

Time evolution generates a canonical transformation: if we consider all possible initial states of a Hamiltonian system and follow all the trajectories for the same time interval, then the map from the initial state to the final state of each trajectory is a canonical transformation. Hamilton–Jacobi theory gives a mixed-variable generating function that generates this time-evolution transformation. For the few integrable systems for which we can solve the Hamilton–Jacobi equation this transformation gets us actionangle coordinates, which form a starting point to study perturbations. If we could find a canonical transformation so that the transformed Hamiltonian was identically