# Existence of a Variational Principle for PDEs with Symmetries and Current Conservation

## TL;DR

This paper establishes a reverse connection between symmetries, conservation laws, and variational principles for PDEs, showing that certain symmetric PDEs can be derived from a variational functional.

## Contribution

It proves that PDEs with sufficient symmetries and conservation laws can be obtained from a variational principle, reversing the traditional Noether theorem perspective.

## Key findings

- PDEs with symmetries lead to conservation laws
- Such PDEs can be derived from a variational functional
- The reverse of Noether's theorem is established

## Abstract

We prove that under certain assumptions a partial differential equation can be derived from a variational principle. It is well-known from Noether's theorem that symmetries of a variational functional lead to conservation laws of the corresponding Euler-Lagrange equation. We reverse this statement and prove that a differential equation which satisfies sufficiently many symmetries and corresponding conservation laws leads to a variational functional whose Euler-Lagrange equation is the given differential equation.