"Hamiltonian mechanics on Lie algebroid"

Abstract: The traditional setting of classical mechanics is usually a Lagrangian function LL on a tangent bundle TQTQ which contains the configuration of the system (the configuration space) or a Hamiltonian vector field on a cotangent bundle T∗QT∗Q (or more generally, a symplectic manifold) which is called the phase space and can be obtained from the Lagrangian formulation through Legendre transform if the the Lagrangian is hyperregular. Nonholonomic mechanics deals with constraints which are not integrable or don't satisfy submanifold condition. Courant {Courant} studied a special kind of vector subbundle called Dirac structure on the direct sum of a vector space and its dual, or more generally, on the Pontryagin bundle of a smooth manifold. This Dirac structure is used in Yoshimura and Marsden {YoMa1} to describe nonholonomic mechanics on tangent bundle. A variational formulation is proposed in {YoMa2} to generalize d'Alembert principle.

Our work aims to generalize these result from tangent bundle to Lie algebroid, which is a generalization of Lie algebra and tangent bundle.

Host: Ari Stern