{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:45:29Z","timestamp":1760240729572,"version":"build-2065373602"},"reference-count":23,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2019,8,5]],"date-time":"2019-08-05T00:00:00Z","timestamp":1564963200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>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\u2013Lagrange) 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.<\/jats:p>","DOI":"10.3390\/sym11081008","type":"journal-article","created":{"date-parts":[[2019,8,5]],"date-time":"2019-08-05T11:17:47Z","timestamp":1565003867000},"page":"1008","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Quasi-Noether Systems and Quasi-Lagrangians"],"prefix":"10.3390","volume":"11","author":[{"given":"V.","family":"Rosenhaus","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, California State University, Chico, CA 95929, USA"}]},{"given":"Ravi","family":"Shankar","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Washington, Seattle, WA 98195, USA"}]}],"member":"1968","published-online":{"date-parts":[[2019,8,5]]},"reference":[{"key":"ref_1","first-page":"235","article-title":"Invariantevariationsprobleme, Nachr. K\u00f6nig. Gessell. Wissen. G\u00f6ttingen","volume":"K1","author":"Noether","year":"1918","journal-title":"Math. Phys."},{"unstructured":"Olver, P.J. (2000). Applications of Lie Groups to Differential Equations, Springer Science & Business Media.","key":"ref_2"},{"unstructured":"Rosen, J. (1972). Some Properties of the Euler\u2013Lagrange Operators, Tel-Aviv University. Preprint TAUP-269-72.","key":"ref_3"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1998","DOI":"10.1063\/1.530533","article-title":"On symmetries, conservation laws, and variational problems for partial differential equations","volume":"35","author":"Rosenhaus","year":"1994","journal-title":"J. Math. Phys."},{"unstructured":"Rosenhaus, V., and Katzin, G.H. (July, January 29). Noether operator relation and conservation laws for partial differential equations. Proceedings of the First Workshop: Nonlinear Physics: Theory and Experiment, Gallipoli, Italy.","key":"ref_5"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"175205","DOI":"10.1088\/1751-8113\/49\/17\/175205","article-title":"Second Noether theorem for quasi-Noether systems","volume":"49","author":"Rosenhaus","year":"2016","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1514","DOI":"10.1134\/S0040577918100082","article-title":"Sub-symmetries, and their properties","volume":"197","author":"Rosenhaus","year":"2018","journal-title":"Theor. Math. Phys."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1016\/S0034-4877(19)30021-7","article-title":"Sub-Symmetries and Conservation Laws","volume":"83","author":"Rosenhaus","year":"2019","journal-title":"Rep. Math. Phys."},{"doi-asserted-by":"crossref","unstructured":"Ibragimov, N.H. (1985). Transformation Groups Applied to Mathematical Physics, D. Reidel Publishing Co.","key":"ref_9","DOI":"10.1007\/978-94-009-5243-0"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"816","DOI":"10.1007\/BF01017679","article-title":"An analogue of the Noether theorem for non-Noether and nonlocal symmetries","volume":"84","author":"Lunev","year":"1991","journal-title":"Theor. Math. Phys."},{"doi-asserted-by":"crossref","unstructured":"Anco, S. (2017). On the incompleteness of Ibragimov\u2019s conservation law theorem and its equivalence to a standard formula using symmetries and adjoint-symmetries. Symmetry, 9.","key":"ref_11","DOI":"10.3390\/sym9030033"},{"unstructured":"Rosenhaus, V. (1996, January 15\u201320). Differential Identities, Symmetries, Conservation Laws and Inverse Variational Problems. Proceedings of the XXI Intern. Colloquium on Group Theoretical Methods in Physics, Goslar, Germany.","key":"ref_12"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1365","DOI":"10.1088\/0305-4470\/20\/6\/020","article-title":"Pseudo-symmetries, Noether\u2019s theorem and the adjoint equation","volume":"20","author":"Sarlet","year":"1987","journal-title":"J. Phys. A"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1007\/BF01405491","article-title":"Local Symmetries and Conservation Laws","volume":"2","author":"Vinogradov","year":"1984","journal-title":"Acta Appl. Math."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"2869","DOI":"10.1103\/PhysRevLett.78.2869","article-title":"Direct construction of conservation laws from field equations","volume":"78","author":"Anco","year":"1997","journal-title":"Phys. Rev. Lett."},{"key":"ref_16","first-page":"534","article-title":"Closed forms associated with linear differential operators","volume":"16","author":"Vladimirov","year":"1980","journal-title":"Differ. Equ."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"745","DOI":"10.1007\/BF01035536","article-title":"Conservation laws of evolution systems","volume":"68","author":"Zharinov","year":"1986","journal-title":"Theor. Math. Phys."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"973","DOI":"10.1063\/1.527117","article-title":"Symmetry transformations, isovectors, and conservation laws","volume":"27","author":"Caviglia","year":"1986","journal-title":"J. Math. Phys."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"432002","DOI":"10.1088\/1751-8113\/44\/43\/432002","article-title":"Nonlinear self-adjointness and conservation laws","volume":"44","author":"Ibragimov","year":"2011","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"106","DOI":"10.2991\/jnmp.2002.9.s1.10","article-title":"Adjoint symmetry constraints leading to binary nonlinearization","volume":"9","author":"Ma","year":"2002","journal-title":"J. Nonlinear Math. Phys."},{"doi-asserted-by":"crossref","unstructured":"Anco, S., Rosa, M., and Gandarias, M. (2017). Conservation laws and symmetries of time-dependent generalized KdV equations. arXiv.","key":"ref_21","DOI":"10.3934\/dcdss.2018035"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"459","DOI":"10.1016\/j.jmaa.2005.02.007","article-title":"Conservation laws for nonlinear telegraph equations","volume":"310","author":"Bluman","year":"2005","journal-title":"J. Math. Anal. Appl."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"604","DOI":"10.1080\/14029251.2019.1640470","article-title":"Variational Operators, Symplectic Operators, and the Cohomology of Scalar Evolution Equations","volume":"26","author":"Fels","year":"2019","journal-title":"J. Nonlinear Math. 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