A universal solution

TL;DR

This paper introduces a universal class of solutions to second order PDEs derived from variational principles, emphasizing Lorentz invariance and the uniqueness of associated Lagrangians, extending to multiple fields.

Contribution

It presents a novel universal solution framework for second order PDEs from homogeneous Lagrangians, highlighting Lorentz invariance's role in ensuring uniqueness.

Findings

Universal solutions solve a wide class of PDEs from variational principles.

Lorentz invariance constrains Lagrangians to a unique form.

Extension of solutions to multiple fields and their equivalence to Companion Lagrangians.

Abstract

The phenomenon of an implicit function which solves a large set of second order partial differential equations obtainable from a variational principle is explicated by the introduction of a class of universal solutions to the equations derivable from an arbitrary Lagrangian which is homogeneous of weight one in the field derivatives. This result is extended to many fields. The imposition of Lorentz invariance makes such Lagrangians unique, and equivalent to the Companion Lagrangians introduced in [baker].