Multisymplectic and polysymplectic structures on fiber bundles

TL;DR

This paper introduces multisymplectic and polysymplectic structures on fiber bundles, defines the symbol of a multisymplectic form, and proves Darboux theorems for local canonical coordinates.

Contribution

It develops a general framework for multisymplectic and polysymplectic geometry on fiber bundles, including the concept of the symbol and Darboux theorems.

Findings

Defined the symbol of a multisymplectic form as a leading order polysymplectic form

Proved Darboux theorems ensuring local canonical coordinates exist

Established foundational structures for multisymplectic geometry on fiber bundles

Abstract

We introduce the concepts of a multisymplectic structure and a polysymplectic structure on a general fiber bundle over a general base manifold, define the concept of the symbol of a multisymplectic form, which is a polysymplectic form representing its leading order contribution, and prove Darboux theorems for the existence of canonical local coordinates.