Optimal Oscillation - Center Transformations

Systems with Hamiltonians of the form Ho(p) + Hi(q, p, t) are considered. A variational principle is proposed for defining that canonical transformation, continuously connected with the identity transformation, which minimizes the residual, coordinate-dependent part of the new Hamiltonian. The principle is based on minimization of the mean square generalized force over phase space and time. The transformation reduces to the action-angle transformation in that part of the phase space of an integrable system where the orbit topology is that of the unperturbed system, or on primary KAM surfaces. General arguments in favour of this definition are given, based on Galilean invariance, decay of the Fourier spectrum, and its ability to include external fields or inhomogeneous systems. The optimal oscillation-center transformation for the physical pendulum (or particle in a sinusoidal potential) is constructed analytically. A modified principle for relativistic systems is presented in an appendix.