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# Poisson Brackets with Prescribed Casimirs

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http://dx.doi.org/10.4153/CJM-2011-082-2
Canad. J. Math. 64(2012), 991-1018
Published:2011-11-15
Printed: Oct 2012
• Pantelis A. Damianou,
Department of Mathematics and Statistics, University of Cyprus, P.O. Box 20537, 1678 Nicosia, Cyprus
• Fani Petalidou,
Department of Mathematics and Statistics, University of Cyprus, P.O. Box 20537, 1678 Nicosia, Cyprus
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## Abstract

We consider the problem of constructing Poisson brackets on smooth manifolds $M$ with prescribed Casimir functions. If $M$ is of even dimension, we achieve our construction by considering a suitable almost symplectic structure on $M$, while, in the case where $M$ is of odd dimension, our objective is achieved by using a convenient almost cosymplectic structure. Several examples and applications are presented.

Keywords:  Poisson bracket, Casimir function, almost symplectic structure, almost cosymplectic structure

MSC Classifications:  53D17 - Poisson manifolds; Poisson groupoids and algebroids 53D15 - Almost contact and almost symplectic manifolds
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