Internal Lagrangians of PDEs as variational principles

TL;DR

This paper explores the intrinsic geometric structure of variational equations in PDEs, introducing internal Lagrangians, and reformulating Noether's theorem to deepen understanding of symmetries and conservation laws.

Contribution

It introduces the concept of internal Lagrangians and establishes their connection with symmetries, conservation laws, and Noether's theorem within the geometric framework of PDEs.

Findings

Internal Lagrangians reproduce stationary action principles.

A new formulation of Noether's theorem using internal Lagrangians.

Examples illustrating the relation between Lagrangians and internal Lagrangians.

Abstract

A description of how the principle of stationary action reproduces itself in terms of the intrinsic geometry of variational equations is proposed. A notion of stationary points of an internal Lagrangian is introduced. A connection between symmetries, conservation laws and internal Lagrangians is established. Noether's theorem is formulated in terms of internal Lagrangians. A relation between non-degenerate Lagrangians and the corresponding internal Lagrangians is investigated. Several examples are discussed.