Bi-presymplectic chains of co-rank one and related Liouville integrable systems

TL;DR

This paper explores bi-presymplectic chains of co-rank one, establishing conditions for their relation to Liouville integrable systems and developing an algorithmic method for constructing separation coordinates.

Contribution

It introduces the notion of dual Poisson-presymplectic pairs and d-compatibility, providing new insights into bi-presymplectic and bi-Hamiltonian chains for integrable systems.

Findings

Conditions for bi-presymplectic chains to represent Liouville integrable systems

Algorithmic construction of separation coordinates from bi-presymplectic representation

Detailed examples of bi-presymplectic and bi-Hamiltonian chains in ${\mathbb R}^3$

Abstract

Bi-presymplectic chains of one-forms of co-rank one 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 derived. To present the construction of bi-presymplectic chains, the notion of dual Poisson-presymplectic pair is used and the concept of d-compatibility of Poisson bivectors and d-compatibility of presymplectic forms is introduced. It is shown that bi-presymplectic representation of related flow leads directly to the construction of separation coordinates in purely algorithmic way. As an illustration bi-presymplectic and bi-Hamiltonian chains in ${\mathbb R}^3$ are considered in detail.