m at h . D G ] 1 3 A ug 2 00 5 1 DYNAMICAL SYSTEMS ON LEIBNIZ ALGEBROIDS

In this paper we study the differential systems on Leibniz algebroids. We introduce a class of almost metriplectic manifolds as a special case of Leibniz manifolds. Also, the notion of almost metriplectic algebroid is introduced. These types of algebroids are used in the presentation of associated differential systems. We give some interesting examples of differential systems on algebroids and the orbits of the solutions of corresponding systems are established. 1 Introduction Lie algebroids have been introduced repeatedly into differential geometry since the early 1950's, and also into physics and algebra, under a wide variety of names. It is well known that there exists a one-to-one correspondence between Lie algebroid structures on a vector bundle π : E → M and linear Poisson structures on the dual vector bundle π * : E * → M. This correspondence can be extended to much wider class of binary operations (brackets) on sections of π on one side, and linear 2-contravariant tensor field on E * on the other side. It is not necessary for these operations to be antisymmetric or to satisfy the Jacoby identity. The vector bundle π togheter with a bracket operation, or the equivalent 2-contravariant tensor field, will be called an algebroid. We mention the concept of Loday algebras, i.e. Leibniz algebras in the sense of Loday which are " non antisymmetric Lie algebras ". Weinstein 's paper on Lagrangian mechanics and groupoids (see [6]) roused new interest into the field of algebroids and groupoids. Weinstein introduces " Lagrangian systems " on a Lie algebroid by means of a Legendre-type map from E to E * associated to a given function L on E. The local coordinate expression of such equations reads ˙ x i = ρ i a (x)y a 1 2000 Mathematics Subject Classification: 17B66, 53C15, 58F05.