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On a manifold equipped with a bivector field, we introduce for everyHamiltonian a Lagrangian on paths valued in the cotangent space whosestationary points projects onto Hamiltonian vector fields. We show that theremaining components of those stationary points tell whether the bivector fieldis Poisson or at least defines an integrable distribution - a class of bivectorfields generalizing twisted Poisson structures that we study in detail.

A Lagrangian for Hamiltonian vector fields on singular Poisson manifolds