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Open AccessArticle

Quasi-Noether Systems and Quasi-Lagrangians

Department of Mathematics and Statistics, California State University, Chico, CA 95929, USA
Department of Mathematics, University of Washington, Seattle, WA 98195, USA
Author to whom correspondence should be addressed.
Symmetry 2019, 11(8), 1008;
Received: 5 July 2019 / Revised: 1 August 2019 / Accepted: 2 August 2019 / Published: 5 August 2019
(This article belongs to the Special Issue Noether's Theorem and Symmetry)
PDF [292 KB, uploaded 5 August 2019]


We study differential systems for which it is possible to establish a correspondence between symmetries and conservation laws based on Noether identity: quasi-Noether systems. We analyze Noether identity and show that it leads to the same conservation laws as Lagrange (Green–Lagrange) identity. We discuss quasi-Noether systems, and some of their properties, and generate classes of quasi-Noether differential equations of the second order. We next introduce a more general version of quasi-Lagrangians which allows us to extend Noether theorem. Here, variational symmetries are only sub-symmetries, not true symmetries. We finally introduce the critical point condition for evolution equations with a conserved integral, demonstrate examples of its compatibility, and compare the invariant submanifolds of quasi-Lagrangian systems with those of Hamiltonian systems. View Full-Text
Keywords: symmetries; conservation laws; Noether operator identity; quasi-Noether systems; quasi-Lagrangians symmetries; conservation laws; Noether operator identity; quasi-Noether systems; quasi-Lagrangians
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).

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Rosenhaus, V.; Shankar, R. Quasi-Noether Systems and Quasi-Lagrangians. Symmetry 2019, 11, 1008.

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