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.