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Bi‐Presymplectic Representation of Liouville Integrable Systems and Related Separability Theory
Author(s) -
Błaszak Maciej
Publication year - 2011
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1111/j.1467-9590.2010.00505.x
Subject(s) - integrable system , mathematics , compatibility (geochemistry) , poisson distribution , pure mathematics , hamiltonian system , mathematical analysis , statistics , geochemistry , geology
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. 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 Stäckel 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.