Bi‐Presymplectic Representation of Liouville Integrable Systems and Related Separability Theory

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 bipresymplectic 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 Stäckel separable systems have biinverse-Hamiltonian representation, i.e. are represented by bi-presymplectic chains of closed oneforms. The co-rank of related structures depends on the explicit form of separation relations.