On a geometric framework for Lagrangian supermechanics

TL;DR

This paper develops a geometric framework for Lagrangian supermechanics using supermanifolds, extending classical mechanics to include both commuting and anticommuting variables within a categorical approach.

Contribution

It introduces a novel geometric formulation of supermechanics that generalizes traditional phase space dynamics to supermanifolds without fixing Grassmann algebra parametrizations.

Findings

Defines phase dynamics as an implicit differential equation in supermanifolds.

Employs categorical approach to supermanifolds for generality.

Provides a geometric foundation for supermechanics.

Abstract

We re-examine classical mechanics with both commuting and anticommuting degrees of freedom. We do this by defining the phase dynamics of a general Lagrangian system as an implicit differential equation in the spirit of Tulczyjew. Rather than parametrising our basic degrees of freedom by a specified Grassmann algebra, we use arbitrary supermanifolds by following the categorical approach to supermanifolds.