Lie Algebroids and Mechanics

We give a formulation of certain types of mechanical systems using the structure of groupoid of the tangent and cotangent bundles to the connguration manifold M; the set of units is the zero section identiied with the manifold M. We study the Legendre transformation on Lie algebroids. The purpose of this paper is not to prove new results but to investigate certain elementary tools utilized in Analytical Mechanics, mainly in C. Marle's paper M] and in A. Weinstein's paper W]. Some ideas are suggested by the papers of P. Dazord D] and I. Vaisman V]. The motion of a point in a Euclidian space can be described by the solution of the Newton equation m d 2 x dt 2 = F ; where d 2 x dt 2 represents the acceleration and F the force; with suitable unities we may suppose that m = 1. When studying more general mechanical systems (especially constrained mechanical systems) it is more diicult to deene the acceleration and the forces. All authors (see for instance A], A.M], G], H], S]) do not agree on the deenitions. For instance the constraint force is a vector eld or a Pfaaan form; there is also an ambiguity about the origin of vectors and covectors. To unify the terminology we have used the groupoid structure of the bundles TM and T M. Each ber is endowed with an aane structure. The set of units is the zero section identiied with the manifold M. The Lie algebroid of these groupoids is the vertical bundle along the zero section. We distinguish vectors and covectors whose origin lies in the zero section from elements whose origin lies in the bers (called lifted elements).