Geometric Hamiltonian Formalism for Reparametrization Invariant Theories with Higher Derivatives

TL;DR

This paper develops a geometric Hamiltonian formalism for reparametrization invariant theories with higher derivatives, revealing the structure of the phase space and Hamiltonian equations using advanced geometric tools.

Contribution

It introduces a novel geometric approach to formulating Hamiltonian dynamics for higher-derivative, reparametrization invariant theories, including a new construction of the phase bundle and Hamilton equations.

Findings

The phase bundle is always odd-dimensional.

The canonical symplectic form induces a null-direction field on the phase bundle.

The Hamilton equations are expressed via a generalized Nambu bracket.

Abstract

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.