Sub-Symmetries and Conservation Laws

V. Rosenhaus, Ravi Shankar

Research output: Contribution to journalArticlepeer-review

1 Scopus citations


The concept of sub-symmetry of a differential system was introduced in [40], where it was shown that a sub-symmetry is a considerably more powerful tool than a regular symmetry with regard to deformation of conservation laws. In this paper, we study the nature of a correspondence between sub-symmetries and conservation laws of a differential system. We show that for a large class of non-Lagrangian systems, there is a natural association between sub-symmetries and local conservation laws based on the Noether operator identity, and we prove an analogue of the first Noether theorem for sub-symmetries. We also demonstrate that a similar association can be established for infinite sub-symmetries of the system. We discuss the role of sub-symmetries in generation of infinite series of conservation laws for the constrained Maxwell system and the incompressible Euler equations of fluid dynamics. Despite the fact that infinite symmetries (with arbitrary functions of dependent variables) are not known for the Euler equations, we find infinite sub-symmetries for the two- and three-dimensional Euler equations with certain constraints. We show that these sub-symmetries generate known series of infinite conservation laws, and obtain new classes of infinite conservation laws.

Original languageEnglish (US)
Pages (from-to)21-48
Number of pages28
JournalReports on Mathematical Physics
Issue number1
StatePublished - Mar 2019
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics


  • Euler equations
  • infinite conservation laws
  • Maxwell's equations
  • non-Lagrangian systems
  • symmetry properties


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