Sprays and Dirac Structures

TL;DR

This paper explores the conditions under which sprays can be described using Hamiltonian mechanics via Dirac structures, introducing methods to find constants of motion and extend to semi-sprays.

Contribution

It proposes a novel approach using gauge transformations and almost Dirac structures to establish Hamiltonian descriptions and identify constants of motion for sprays.

Findings

Established conditions for Hamiltonian description of sprays

Developed a method to find constants of motion using gauge transformations

Extended approach to semi-sprays with modifications

Abstract

We look at the the possibility of a spray having a Hamiltonian description considering Dirac structure as underlying geometric structure. In a simpler scenario, we consider almost Dirac structure as an auxiliary object to find constants of motion for sprays. Staring with the horizontal distribution associated to a spray, we use gauge transformations generated by two forms in order to obtain constants of motion and possible Hamiltonian description. We also apply our approach to semi-spray with some modifications.