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arXiv · 2009.04102

A modified formal Lagrangian formulation for general differential equations

Linyu Peng
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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.

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Linyu Peng. 2022-02-09. A modified formal Lagrangian formulation for general differential equations. https://doi.org/10.1007/s13160-022-00500-7

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