Hamiltonian semisprays on Lie algebroids

TL;DR

This paper extends the concept of Hamiltonian semisprays from tangent bundles to arbitrary Lie algebroids, constructing a family of Poisson brackets to facilitate second-order Hamiltonian dynamics in this broader setting.

Contribution

It generalizes the existence of Hamiltonian semisprays to Lie algebroids and develops a geometric framework linking Poisson geometry with algebroid structures.

Findings

Constructed a family of Poisson brackets on Lie algebroids.

Extended classical Hamiltonian dynamics to Lie algebroids.

Revealed new interactions between Poisson geometry and algebroid structures.

Abstract

We study the existence of Hamiltonian semisprays on Lie algebroids. This work is motivated by a problem studied by Vaisman for tangent bundles, and we extend this question to the setting of arbitrary Lie algebroids and provide a general solution. More precisely, given a Lie algebroid and a regular Lagrangian, we construct a family of Poisson brackets on the algebroid such that the Hamiltonian vector field associated with the corresponding energy function is a semispray. Our approach is based on the symplectic geometry of the prolongation of a Lie algebroid and a cohomological analysis of its vertical subbundle. The results provide a geometric framework for second-order Hamiltonian dynamics on Lie algebroids, extending some known facts in the classical tangent bundle case and revealing new interactions between Poisson geometry and algebroid structures.