A multiple scales approach to maximal superintegrability

Abstract

In this paper we present a simple, algorithmic test to establish if a Hamiltonian system is maximally superintegrable or not. This test is based on a very simple corollary of a theorem due to Nekhoroshev and on a perturbative technique called multiple scales method. If the outcome is positive, this test can be used to suggest maximal superintegrability, whereas when the outcome is negative it can be used to disprove it. This method can be regarded as a finite dimensional analog of the multiple scales method as a way to produce soliton equations. We use this technique to show that the real counterpart of a mechanical system found by Jules Drach in 1935 is, in general, not maximally superintegrable. We give some hints on how this approach could be applied to classify maximally superintegrable systems by presenting a direct proof of the well-known Bertrand's theorem.

30 pages, 4 figure

References in corpus (2)

• Reduction of superintegrable systems: the anisotropic harmonic oscillator
• Superintegrability of the Caged Anisotropic Oscillator

Cited by in corpus (4)

• On superintegrability of 3D axially-symmetric non-subgroup-type systems with magnetic fields
• The ${\mathfrak{sl}}_2(\mathbb{R})$ coalgebra symmetry and the superintegrable discrete-time systems
• Superintegrability of 3-dimensional Hamiltonian systems with conformally Euclidean metrics. Oscillator-related and Kepler-related systems
• Superintegrability on the 3-dimensional spaces with curvature. Oscillator-related and Kepler-related systems on the Sphere $S^3$ and on the Hyperbolic space $H^3$