An invitation to multisymplectic geometry

TL;DR

This paper explores the foundational aspects of multisymplectic geometry, including its relation to classical field theories, normal forms, and symmetries, highlighting recent theoretical developments.

Contribution

It provides a comprehensive reformulation of classical field theories in multisymplectic terms and investigates the existence of Darboux-type theorems and symmetry properties.

Findings

Reformulation of Lagrangian and Hamiltonian field theories in multisymplectic geometry

Analysis of normal forms and conditions for Darboux-type theorems

Survey of recent advances in symmetries and conserved quantities

Abstract

In this article we study multisymplectic geometry, i.e., the geometry of manifolds with a non-degenerate, closed differential form. First we describe the transition from Lagrangian to Hamiltonian classical field theories, and then we reformulate the latter in multisymplectic terms. Furthermore, we investigate basic questions on normal forms of multisymplectic manifolds, notably the questions wether and when Darboux-type theorems hold, and how many diffeomorphisms certain, important classes of multisymplectic manifolds possess. Finally, we survey recent advances in the area of symmetries and conserved quantities on multisymplectic manifolds.