Symmetries and gauge symmetries in multisymplectic first and second-order Lagrangian field theories: electromagnetic and gravitational fields

TL;DR

This paper explores the geometric structure of symmetries and conservation laws in multisymplectic formulations of classical field theories, focusing on electromagnetic and gravitational fields, and clarifies the role of gauge symmetries.

Contribution

It provides a geometric framework for understanding Noether and gauge symmetries in multisymplectic first and second-order Lagrangian field theories, with applications to electromagnetism and gravity.

Findings

Noether theorems are formulated in a multisymplectic geometric setting.

Gauge symmetries are characterized geometrically within this framework.

Applications include analysis of electromagnetic and gravitational theories, including Einstein-Hilbert and Einstein-Palatini formulations.

Abstract

Symmetries and, in particular, Cartan (Noether) symmetries and conserved quantities (conservation laws) are studied for the multisymplectic formulation of first and second order Lagrangian classical field theories. Noether-type theorems are stated in this geometric framework. The concept of gauge symmetry and its geometrical meaning are also discussed in this formulation. The results are applied to study Noether and gauge symmetries for the multisymplectic description of the electromagnetic and the gravitational theory; in particular, the Einstein--Hilbert and the Einstein--Palatini approaches.