Higher order corrections to adiabatic invariants of generalized slow-fast Hamiltonian systems

Abstract

We present a coordinate-free approach for constructing approximate first integrals of generalized slow-fast Hamiltonian systems, based on the global averaging method on parameter-dependent phase spaces with \documentclass[12pt]{minimal}\begin{document}$\mathbb {S}^1$\end{document}S1-symmetry. Explicit global formulas for approximate second-order first integrals are derived. As examples, we analyze the case quadratic in the fast variables (in particular, the elastic pendulum), and the charged particle in a slowly-varying magnetic field.