Poisson brackets with prescribed Casimirs

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.