Tangent lifts of bi-Hamiltonian structures

We construct several Poisson structures on the tangent bundle $TM$ to a Poisson manifold $M$ using the Lie algebroid structure on the cotangent bundle $T^*M$. We also show that bi-Hamiltonian structure from $M$ can be transferred to its tangent bundle $TM$. Moreover, we present how to find Casimir functions for those Poisson structures and we discuss some particular examples.