Generalized and graded geometry for mechanics: a comprehensive introduction

TL;DR

This paper reviews recent advances in differential geometry, like higher structures and graded manifolds, and explores their applications in mechanics, emphasizing the use of Q-structures and integrators for improved modeling.

Contribution

It provides a comprehensive overview connecting modern geometric concepts with classical mechanics problems, highlighting the potential of Q-structures in mechanics modeling and numerical integration.

Findings

Q-structures unify geometric constructions in mechanics

Potential of Q-structure preserving integrators in mechanics

Open problems in applying graded geometry to mechanics

Abstract

In this paper we make an overview of results relating the recent "discoveries" in differential geometry, such as higher structures and differential graded manifolds with some natural problems coming from mechanics. We explain that a lot of classical differential geometric constructions in the context can be conveniently described using the language of Q-structures, and thus Q-structure preserving integrators are potentially of great use in mechanics. We give some hints how the latter can be constructed, and formulate some open problems. Since the text is intended both to mathematics and mechanics communities, we tried to make it accessible to non-geometers as well.