A modified formal Lagrangian formulation for general differential equations

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)573-598
Number of pages26
JournalJapan Journal of Industrial and Applied Mathematics
Issue number2
Publication statusPublished - 2022 May


  • Conservation laws
  • Modified formal Lagrangians
  • Noether’s Theorem
  • Self-adjointness
  • Symmetries

ASJC Scopus subject areas

  • Engineering(all)
  • Applied Mathematics


Dive into the research topics of 'A modified formal Lagrangian formulation for general differential equations'. Together they form a unique fingerprint.

Cite this