Bi-presymplectic representation of Liouville integrable systems and related separability theory

TL;DR

This paper introduces a bi-presymplectic framework for Liouville integrable systems, providing conditions for their representation, and demonstrates an algorithmic method for constructing separation coordinates, linking separability to bi-presymplectic chains.

Contribution

It develops a novel bi-presymplectic approach to integrable systems, establishing conditions for their representation and linking separability with bi-presymplectic chains.

Findings

Bi-presymplectic chains can represent Liouville integrable systems.

Separation coordinates can be constructed algorithmically.

Stäckel separable systems have bi-presymplectic representations.

Abstract

Bi-presymplectic chains of one-forms of arbitrary co-rank are considered. The conditions in which such chains represent some Liouville integrable systems and the conditions in which there exist related bi-Hamiltonian chains of vector fields are presented. In order to derived the construction of bi-presymplectic chains, the notions of dual Poisson-presymplectic pair, d-compatibility of presymplectic forms and d-compatibility of Poisson bivectors is used. The completely algorithmic construction of separation coordinates is demonstrated. It is also proved that St\"{a}ckel separable systems have bi-inverse-Hamiltonian representation, i.e. are represented by bi-presymplectic chains of closed one-forms. The co-rank of related structures depends on the explicit form of separation relations.