  The application of the Legendre transformation to a hyperregular Lagrangian
system results in a Hamiltonian vector field generated by a Hamiltonian defined
on the phase space of the mechanical system. The Legendre transformation in its
usual interpretation can not be applied to homogeneous Lagrangians found in
relativistic mechanics. The dynamics of relativistic systems must be formulated
in terms of implicit differential equations in the phase space and not in terms
of Hamiltonian vector fields. The constrained Hamiltonian systems introduced by
Dirac [1] are not general enough to cover some important cases. We formulate a
geometric framework which permits Lagrangian and Hamiltonian descriptions of
the dynamics of a wide class of mechanical systems. Lagrangians and
Hamiltonians are presented as families of functions. The Legendre
transformation and the inverse Legendre transformation are described as
transitions between these families. Two examples, the dynamics of a
relativistic particle and a space-time formulation of geometric optics
(relativistic massless particle), are given.
