Decomposition of scalar potentials of natural Hamiltonians into integrable and perturbative terms: a naive approach

Abstract

The geometry of Stäckel systems is employed to find, for a given (classical) natural Hamiltonian, another natural Hamiltonian which is integrable by separation of variables and in some sense close to the first one. The study is intended as a first step for a perturbative analysis of a given Hamiltonian system. The method proposed here, still in development, is in large part coordinate independent and effective mainly on manifolds of constant curvature. Examples are given for the quadrupole field and Hénon-Heiles systems. Stäckel systems are associated with quadratic in the momenta first integrals which play a fundamental role in quantization of classical systems.