Nonholonomic Algebroids, Finsler Geometry, and Lagrange–Hamilton Spaces

We elaborate an unified geometric approach to classical mechanics, Riemann–Finsler spaces and gravity theories on Lie algebroids provided with nonlinear connection (N–connection) structure. There are investigated the conditions when the fundamental geometric objects like the anchor, metric and linear connection, almost sympletic and related almost complex structures may be canonically defined by a N–connection induced from a regular Lagrangian (or Hamiltonian), in mechanical models, or by generic off–diagonal metric terms and nonholonomic frames, in gravity theories. Such geometric constructions are modelled on nonholonomic manifolds provided with nonintegrable distributions and related chains of exact sequences of submanifolds defining N–connections. We investigate the main properties of the Lagrange, Hamilton, Finsler–Riemann and Einstein–Cartan algebroids and construct and analyze exact solutions describing such objects.