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.
|ジャーナル||Japan Journal of Industrial and Applied Mathematics|
|出版ステータス||Published - 2022 5月|
ASJC Scopus subject areas