Symmetric Divergence-free tensors in the calculus of variations

TL;DR

This paper explores the occurrence of divergence-free symmetric tensors in models governed by the second variational principle, linking them to Euler--Lagrange equations and symmetry invariance in mathematical physics.

Contribution

It reveals that divergence-free symmetric tensors naturally arise in the second variational principle involving closed differential forms, connecting them to fundamental equations and symmetries.

Findings

Divergence-free symmetric tensors are linked to the second variational principle.

These tensors correspond to the second form of Euler--Lagrange equations.

Symmetry invariance of the Lagrangian relates to tensor symmetry.

Abstract

Divergence-free symmetric tensors seem ubiquitous in Mathematical Physics. We show that this structure occurs in models that are described by the so-called "second" variational principle, where the argument of the Lagrangian is a closed differential form. Divergence-free tensors are nothing but the second form of the Euler--Lagrange equations. The symmetry is associated with the invariance of the Lagrangian density upon the action of some orthogonal group.