The Geometry of Integrable and Superintegrable Systems

TL;DR

This paper explores the geometric structures underlying integrable and superintegrable systems, focusing on automorphisms, normal forms, and obstructions to system equivalence, without relying on traditional symplectic or Poisson frameworks.

Contribution

It introduces a geometric framework based on normal forms and generalized toroidal bundles that characterizes integrable systems independently of symplectic or Poisson structures.

Findings

Normal form representation of integrable systems independent of additional structures

Identification of non-canonical diffeomorphisms generating alternative Hamiltonian structures

Energy-period theorem as an obstruction to system equivalence

Abstract

The group of automorphisms of the geometry of an integrable system is considered. The geometrical structure used to obtain it is provided by a normal form representation of integrable systems that do not depend on any additional geometrical structure like symplectic, Poisson, etc. Such geometrical structure provides a generalized toroidal bundle on the carrier space of the system. Non--canonical diffeomorphisms of such structure generate alternative Hamiltonian structures for complete integrable Hamiltonian systems. The energy-period theorem provides the first non--trivial obstruction for the equivalence of integrable systems.