The First-Order Euler-Lagrange equations and some of their uses

TL;DR

This paper discusses a systematic method for reducing the order of nonlinear field equations, such as Euler-Lagrange equations, to find relevant solutions more efficiently, with potential applications in various field theories.

Contribution

It generalizes and develops a specific order reduction method for nonlinear field equations, enhancing its applicability and understanding.

Findings

The method simplifies solving complex nonlinear equations.

It broadens the applicability of order reduction techniques.

Potential for discovering new solutions in field theories.

Abstract

In many nonlinear field theories, relevant solutions may be found by reducing the order of the original Euler-Lagrange equations, e.g., to first order equations (Bogomolnyi equations, self-duality equations, etc.). Here we generalise, further develop and apply one particular method for the order reduction of nonlinear field equations which, despite its systematic and versatile character, is not widely known.