# Noether's Theorem in Multisymplectic Geometry

## TL;DR

This paper generalizes Noether's theorem within multisymplectic geometry, establishing a correspondence between symmetries and conserved quantities, and extends classical momentum concepts to multisymplectic phase spaces, including manifolds with $G_2$ structures.

## Contribution

It introduces a multisymplectic version of Noether's theorem and explores the role of homotopy co-momentum maps, extending classical momentum theory to new geometric contexts.

## Key findings

- Established a correspondence between symmetries and conserved quantities in multisymplectic geometry.
- Generalized classical momentum and position functions to multisymplectic phase spaces.
- Applied the theory to manifolds with torsion-free $G_2$ structures.

## Abstract

We extend Noether's theorem to the setting of multisymplectic geometry by exhibiting a correspondence between conserved quantities and continuous symmetries on a multi-Hamiltonian system. We show that a homotopy co-momentum map interacts with this correspondence in a way analogous to the moment map in symplectic geometry. We apply our results to generalize the theory of the classical momentum and position functions from the phase space of a given physical system to the multisymplectic phase space. We also apply our results to manifolds with a torsion-free $G_2$ structure.