Divergence Invariant Variational Problems*

Peter J. Olver

Research output: Chapter in Book/Report/Conference proceedingChapter

1 Scopus citations


Given a Lie group acting on the space of independent (spacetime) and dependent (field) variables, it is proved that a divergence invariant variational problem is equivalent to a strictly invariant variational problem if and only if a certain associated cohomology class in the invariant variational bicomplex vanishes. This result is illustrated by several examples, starting with the free particle Lagrangian that appeared in Emmy Noether’s original paper, and includes derivations of associated conservation laws through application of her First Theorem. The chapter concludes with some speculations as to the role such cohomology classes might play in fundamental physics, based on the construction of suitable invariant Lagrangians.

Original languageEnglish (US)
Title of host publicationThe Philosophy and Physics of Noether’s Theorems
Subtitle of host publicationA Centenary Volume
PublisherCambridge University Press
Number of pages10
ISBN (Electronic)9781108665445
ISBN (Print)9781108486231
StatePublished - Jan 1 2022

Bibliographical note

Publisher Copyright:
© Cambridge University Press & Assessment 2022.


  • Noether’s theorems
  • variational symmetries


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