Grassmann-graded Lagrangian theory of even and odd variables

TL;DR

This paper develops a comprehensive Grassmann-graded Lagrangian formalism on graded manifolds, capable of describing complex reducible degenerate systems with hierarchies of symmetries and identities, applicable to classical field theory and mechanics.

Contribution

It introduces a general Grassmann-graded variational bicomplex framework that captures reducible degenerate Lagrangian systems with higher-order symmetries.

Findings

Provides a unified formalism for classical field theory and mechanics.

Describes hierarchies of Noether identities and gauge symmetries.

Enables analysis of reducible degenerate Lagrangian systems.

Abstract

Graded Lagrangian formalism in terms of a Grassmann-graded variational bicomplex on graded manifolds is developed in a very general setting. This formalism provides the comprehensive description of reducible degenerate Lagrangian systems, characterized by hierarchies of non-trivial higher-order Noether identities and gauge symmetries. This is a general case of classical field theory and Lagrangian non-relativistic mechanics.