TY - JOUR

T1 - A modified formal Lagrangian formulation for general differential equations

AU - Peng, Linyu

N1 - Funding Information:
This work was partially supported by JSPS KAKENHI Grant Number JP20K14365, JST-CREST Grant Number JPMJCR1914, Keio Gijuku Academic Development Funds, and Keio Gijuku Fukuzawa Memorial Fund. L. Peng is adjunct faculty member at the School of Mathematics and Statistics, Beijing Institute of Technology, China, and adjunct researcher at the Waseda Institute for Advanced Study, Waseda University, Japan. We thank the anonymous reviewers for their valuable comments.
Funding Information:
This work was partially supported by JSPS KAKENHI Grant Number JP20K14365, JST-CREST Grant Number JPMJCR1914, Keio Gijuku Academic Development Funds, and Keio Gijuku Fukuzawa Memorial Fund. L. Peng is adjunct faculty member at the School of Mathematics and Statistics, Beijing Institute of Technology, China, and adjunct researcher at the Waseda Institute for Advanced Study, Waseda University, Japan. We thank the anonymous reviewers for their valuable comments.
Publisher Copyright:
© 2022, The JJIAM Publishing Committee and Springer Japan KK, part of Springer Nature.

PY - 2022/5

Y1 - 2022/5

N2 - 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.

AB - 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.

KW - Conservation laws

KW - Modified formal Lagrangians

KW - Noether’s Theorem

KW - Self-adjointness

KW - Symmetries

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U2 - 10.1007/s13160-022-00500-7

DO - 10.1007/s13160-022-00500-7

M3 - Article

AN - SCOPUS:85124302425

SN - 0916-7005

VL - 39

SP - 573

EP - 598

JO - Japan Journal of Industrial and Applied Mathematics

JF - Japan Journal of Industrial and Applied Mathematics

IS - 2

ER -