A multiple scales approach to maximal superintegrability

TL;DR

This paper introduces an algorithmic test based on multiple scales and Nekhoroshev's theorem to determine if a Hamiltonian system is maximally superintegrable, providing a new tool for classification and analysis.

Contribution

The paper develops a simple, perturbative method using multiple scales to test for maximal superintegrability in Hamiltonian systems, offering a practical and theoretical advancement.

Findings

The test can confirm or disprove maximal superintegrability.

Applied to Drach's system, it shows the system is generally not maximally superintegrable.

Provides a new proof of Bertrand's theorem using this approach.

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.