Lagrangian equations on affine Lie algebroids

We recall the concept of a Lie algebroid on a vector bundle and the associated notion of Lagrange-type equations. A heuristic calculus of variations approach tells us what a time-dependent generalization of such equations should look like. In order to find a geometrical model for such a generalization, the idea of a Lie algebroid structure on a class of affine bundles is introduced. We develop a calculus of forms on sections of such a bundle by looking at its extended dual. It is sketched how the affine Lie algebroid axioms are equivalent to the coboundary property of the exterior derivative in such a calculus. The interest of the new formalism is further illustrated by the fact that one can define a notion of prolongation of the original algebroid. We briefly discuss how this prolongation will provide the key to various geometrical constructions which are the analogues of the well-known geometrical aspects of second-order ordinary differential equations in general, and Lagrangian dynamics in particular.