# How do I do an interaction frame transformation in classical Hamiltonian mechanics?

I am fairly proficient in quantum mechanics but am mostly self taught in classical mechanics (out of Goldstein, Borben and Stehle, and Wikipedia). My question relates to a classical analogue of the interaction picture technique from time-dependent perturbation theory in quantum mechanics. I have studied in detail classical contact transformations and generating functions, but so far have failed to find a general prescription for removing an unwanted term from a Hamiltonian by a frame transformation.

## tl;dr

Consider a Hamiltonian with two components $$H = H_0 + \epsilon H_1$$ where $\epsilon$ is a small multiplicative constant; We consider $\epsilon H_1$ to be a perturbing Hamiltonian. The object is to devise a canonical transformation $(q, p) \rightarrow (Q, P)$ such that

\begin{eqnarray*} \dot{Q} &=& \frac{\partial}{\partial P} K \\ \dot{P} &=& - \frac{\partial}{\partial Q} K \end{eqnarray*} and with $K$ directly proportional to $\epsilon$. If we let $G$ represent the generating function of the canonical transformation, I would think that since \begin{eqnarray} K = H+\frac{\partial }{\partial t}G \end{eqnarray} then we would want $\partial G/ \partial t = - H_0$. Is such a transformation always possible? How do I find the generating function for this transformation?

## Quantum case

I want to remind everyone there is a well-known and general prescription for doing this in quantum mechanics. In the transformed frame, the new Hamiltonian is $$K = U_0^\dagger (H - H_0)U_0 = \epsilon U^\dagger_0 H_1 U_0$$ where the propagator $U_0$ satisfies the Schrodinger equation $i \hbar \dot{U}_0 = H_0 U_0$. It's trivial to calculate the transformed operators for any observable. For example, the transformed phase space coordinate operators are $Q = U_0^\dagger q U_0$ and $P = U_0^\dagger p U_0$. Since the Ehrenfest theorem is a real thing, I would expect the equations of motion for the quantum operators to agree with the classical variables produced by the canonical transformation technique.

Canonical perturbation theory uses action-angle coordinates to describe an integrable Hamiltonian $H_0(J)$ (where $J$ is the action). A perturbation term can then be added such that the total Hamiltonian is non-integrable $$H = H_0 +\epsilon H_1$$ We then have Hamilton's equations $$\dot J = -\epsilon \nabla _Q H_1 ,\qquad \dot Q = \nabla _JH_0 + \epsilon\nabla_JH_1$$ Generating functions can then be used, expanded to desired order and then a Hamilton-Jacobi equation can be written. The Hamiltonian can be truncated and the fundamental equation of Poincaré can be written.