Symmetries of Hamiltonian systems on Lie algebroids

In the present paper, we study infinitesimal symmetries, natural infinitesimal symmetries, Newtonoid sections, infinitesimal Noether symmetries, and conservation laws for Hamiltonian systems within the general framework of Lie algebroids. Using dynamical covariant derivatives and Jacobi endomorphisms, we find invariant equations of some type of symmetries and prove that the canonical nonlinear connection induced by a regular Hamiltonian can be determined by these symmetries. Finally, we present examples from the optimal control theory that prove that the framework of Lie algebroids is more useful than the cotangent bundle in order to study the symmetries for the dynamics induced by a Hamiltonian function.