  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.
