Vol. 9, No. 1, 2021

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Generalized and graded geometry for mechanics: a comprehensive introduction

Vladimir Salnikov, Aziz Hamdouni and Daria Loziienko

Vol. 9 (2021), No. 1, 59–75
DOI: 10.2140/memocs.2021.9.59
Abstract

In this paper we give 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 this 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 this paper is intended for both the mathematics and mechanics communities, we tried to make it accessible to nongeometers as well.

Keywords
Dirac structures, differential graded manifolds, theoretical mechanics, geometric integrators
Mathematical Subject Classification
Primary: 70G45, 58A50, 70H45, 70H30
Milestones
Received: 16 July 2020
Revised: 14 September 2020
Accepted: 29 October 2020
Published: 17 March 2021

Communicated by Francesco dell'Isola
Authors
Vladimir Salnikov
Laboratoire des Sciences de l’Ingénieur pour l’Environnement (LaSIE)
UMR CNRS 7356
La Rochelle Université
La Rochelle
France
Aziz Hamdouni
Laboratoire des Sciences de l’Ingénieur pour l’Environnement (LaSIE)
UMR CNRS 7356
La Rochelle Université
La Rochelle
France
Daria Loziienko
Laboratoire des Sciences de l’Ingénieur pour l’Environnement (LaSIE)
UMR CNRS 7356
La Rochelle Université
La Rochelle
France