A modified formal Lagrangian formulation for general differential equations

TL;DR

This paper introduces a modified formal Lagrangian approach for any differential system, enabling the derivation of conservation laws and symmetries, exemplified through the Fornberg-Whitham and Euler equations.

Contribution

It presents a novel modified formal Lagrangian formulation incorporating dummy variables, applicable to all differential equations, and links conservation laws with variational symmetries.

Findings

Derived a nontrivial conservation law for the Fornberg-Whitham equation.

Established a correspondence between conservation laws and symmetries of Euler equations.

Proved the existence of the modified Lagrangian for any differential system.

Abstract

In this paper, we propose a modified formal Lagrangian formulation by introducing dummy dependent variables and prove the existence of such a formulation for any system of differential equations. The corresponding Euler--Lagrange equations, consisting of the original system and its adjoint system about the dummy variables, reduce to the original system via a simple substitution for the dummy variables. The formulation is applied to study conservation laws of differential equations through Noether's Theorem and in particular, a nontrivial conservation law of the Fornberg--Whitham equation is obtained by using its Lie point symmetries. Finally, a correspondence between conservation laws of the incompressible Euler equations and variational symmetries of the relevant modified formal Lagrangian is shown.