AV-differential geometry: Euler-Lagrange equations

A general, consistent and complete framework for geometrical formulation of mechanical systems is proposed, based on certain structures on affine bundles (affgebroids) that generalize Lie algebras and Lie algebroids. This scheme covers and unifies various geometrical approaches to mechanics in the Lagrangian and Hamiltonian pictures, including time-dependent lagrangians and hamiltonians. In our approach, lagrangians and hamiltonians are, in general, sections of certain R-principal bundles, and the solutions of analogs of Euler-Lagrange equations are curves in certain affine bundles. The correct geometrical and frame-independent description of Newtonian Mechanics is of this type. MSC 2000: 70H99, 17B66, 53D10, 70H03, 70H05, 53C99, 53D17.