  An explicit expression for the Jacobi metric for a general Lagrangian system
is obtained as a series expansion in the square root of the kinetic energy of
the system and the corresponding geodesics are described in terms of an
appropriate non-linear connection and the associated curvature. In the limit of
low kinetic energies the trajectories of motion of any Lagrangian system are
very well approximated by the geodesics of an energy dependent Randers metric
or, equivalently, by the paths in configuration space of a representative point
subject to electromagnetic- and gravitational-like force fields. For higher
kinetic energy values the trajectories of motion are instead the geodesics of a
general energy dependent Finsler metric, corresponding also to the paths in
configuration space of a representative point subject to a hierarchy of a
potentially infinite number of covariant force fields that generalise the
electromagnetic and gravitational ones. Some general implications of these
findings are discussed.
