{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,8]],"date-time":"2025-11-08T13:41:43Z","timestamp":1762609303268,"version":"build-2065373602"},"reference-count":43,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2023,4,24]],"date-time":"2023-04-24T00:00:00Z","timestamp":1682294400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundation of China","award":["12172241","11972241","12272248","11802193"],"award-info":[{"award-number":["12172241","11972241","12272248","11802193"]}]},{"name":"Qing Lan Project of colleges and universities in Jiangsu Province","award":["12172241","11972241","12272248","11802193"],"award-info":[{"award-number":["12172241","11972241","12272248","11802193"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Generalized operators have recently been proposed with great potential applications. Here, we present research carried out on Noether figury and perturbation to Noether symmetry for Hamiltonian systems within generalized operators. There are four parts, and each part contains two kinds of generalized operator. Firstly, Hamilton equations are established. Secondly, the Noether symmetry method is used for finding the solutions to the differential equations of motion, and conserved quantities are obtained. Thirdly, perturbation to Noether symmetry and adiabatic invariants are further explored. In the end, two examples are given to illustrate the methods and results.<\/jats:p>","DOI":"10.3390\/sym15050973","type":"journal-article","created":{"date-parts":[[2023,4,25]],"date-time":"2023-04-25T02:06:35Z","timestamp":1682388395000},"page":"973","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Research on the Symmetry of the Hamiltonian System under Generalized Operators"],"prefix":"10.3390","volume":"15","author":[{"given":"Cai","family":"Wang","sequence":"first","affiliation":[{"name":"School of Mathematical Sciences, Suzhou University of Science and Technology, Suzhou 215009, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4312-8034","authenticated-orcid":false,"given":"Chuan-Jing","family":"Song","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Suzhou University of Science and Technology, Suzhou 215009, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,4,24]]},"reference":[{"key":"ref_1","unstructured":"Zhu, Z.X., Zhou, Q.Z., and Yin, J.S. 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