Geometric Hamiltonian formalism for reparameterization-invariant theories with higher derivatives

We consider reparameterization-invariant Lagrangian theories with higher derivatives, investigate the geometric structures behind these theories, and construct the Hamiltonian formalism geometrically. We present the Legendre transformation formula for such systems, which differs from the usual one. We show that the phase bundle, i.e., the image of the Legendre transformation, is a submanifold of a certain cotangent bundle, and this submanifold is always odd-dimensional in this construction. Therefore, the canonical symplectic 2-form of the ambient cotangent bundle generates a field on the phase bundle of null directions of its restriction. We show that the integral lines of this field project to the extremals of the action on the configuration manifold. This means that the obtained field is a Hamiltonian field. We write the corresponding Hamilton equations in terms of the generalized Nambu bracket.