Nonholonomic algebroids, Finsler geometry, and Lagrange-Hamilton spaces

We elaborate a 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 conditions when the fundamental geometric objects (anchor, metric and linear connection, almost symplectic, and related almost complex structures) may be canonically defined by an 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, construct and analyze exact solutions describing such objects.