Hamiltonian versus Lagrangian formulations of supermechanic

TL;DR

This paper develops intrinsic geometric definitions of key supermechanical objects using graded manifolds, establishing a correspondence between Lagrangian and Hamiltonian formulations in supermechanics.

Contribution

It introduces a geometric framework for supermechanics that unifies Lagrangian and Hamiltonian approaches via graded manifold structures.

Findings

Defined supermechanical objects intrinsically on graded manifolds

Established a correspondence between Lagrangian and Hamiltonian formulations

Provided geometric tools for supermechanics analysis

Abstract

We take advantage of different generalizations of the tangent manifold to the context of graded manifolds, together with the notion of super section along a morphism of graded manifolds, to obtain intrinsic definitions of the main objects in supermechanics such as, the vertical endomorphism, the canonical and the Cartan's graded forms, the total time derivative operator and the super--Legendre transformation. In this way, we obtain a correspondence between the Lagrangian and the Hamiltonian formulations of supermechanics.