A multiple scales approach to maximal superintegrability

• Main Authors: Gubbiotti, G, Latini, D
• Format: Journal Article
• Language: English
• Published: 10-11-2017
• Subjects: Physics - Exactly Solvable and Integrable Systems
• DOI: 10.48550/arxiv.1711.03719

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.