  Reparametrization invariant Lagrangian theories with higher derivatives are
considered. We investigate the geometric structures behind these theories and
construct the Hamiltonian formalism in a geometric way. The Legendre
transformation which corresponds to the transition from the Lagrangian
formalism to the Hamiltonian formalism is non-trivial in this case. The
resulting phase bundle, i.e. the image of the Legendre transformation, is a
submanifold of some cotangent bundle. We show that in our construction it is
always odd-dimensional. Therefore the canonical symplectic two-form from the
ambient cotangent bundle generates on the phase bundle a field of the
null-directions of its restriction. It is shown that the integral lines of this
field project directly to the extremals of the action on the configuration
manifold. Therefore this naturally arising field is what is called the Hamilton
field. We also express the corresponding Hamilton equations through the
generilized Nambu bracket.
