Elliptic Curves and a New Construction of Integrable Systems

TL;DR

This paper introduces a new class of integrable dynamical systems on e(3) derived from elliptic curves and Lax matrices, proving their integrability and providing separation of variables, with connections to the Hess-Appel'rot system.

Contribution

It constructs a novel family of integrable systems on e(3) using elliptic curves and Lax matrices, including non-Hamiltonian cases and separation of variables methods.

Findings

Five integrable cases are identified, with three being Hamiltonian.

All five cases are proven to be integrable.

Separation of variables is achieved in Sklyanin's sense for all cases.

Abstract

A class of elliptic curves with associated Lax matrices is considered. A family of dynamical systems on e(3) parametrized by polynomial a with above Lax matrices are constructed. Five cases from the family are selected by the condition of preserving the standard mea- sure. Three of them are Hamiltonian. It is proved that two other cases are not Hamiltonian in the standard Poisson structure on e(3). Integra- bility of all five cases is proven. Integration procedures are performed in all five cases. Separation of variables in Sklyanin sense is also given. A connection with Hess-Appel'rot system is established. A sort of sep- aration of variables is suggested for the Hess-Appel'rot system.