Lagrangian Formalism Over Graded Algebras

TL;DR

This paper develops an algebraic framework for the Lagrangian formalism over graded algebras, laying groundwork for noncommutative variational calculus and related conservation laws.

Contribution

It introduces a noncommutative algebraic setting for Lagrangian formalism, including integration, adjoint operators, Green's formula, and Noether's theorem, advancing noncommutative variational theory.

Findings

Established a noncommutative integration procedure

Defined adjoint operators and Green's formula in this setting

Connected conservation laws with algebraic structures

Abstract

This paper provides a description of an algebraic setting for the Lagrangian formalism over graded algebras and is intended as the necessary first step towards the noncommutative C-spectral sequence (variational bicomplex). A noncommutative version of integration procedure, the notion of adjoint operator, Green's formula, the relation between integral and differential forms, conservation laws, Euler operator, Noether's theorem is considered.