{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,8]],"date-time":"2026-02-08T19:47:11Z","timestamp":1770580031105,"version":"3.49.0"},"reference-count":15,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2021,7,1]],"date-time":"2021-07-01T00:00:00Z","timestamp":1625097600000},"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>A natural example of evolution can be described by a time-dependent two degrees-of-freedom Hamiltonian. We choose the case where initially the Hamiltonian derives from a general cubic potential, the linearised system has frequencies 1 and \u03c9&gt;0. The time-dependence produces slow evolution to discrete (mirror) symmetry in one of the degrees-of-freedom. This changes the dynamics drastically depending on the frequency ratio \u03c9 and the timescale of evolution. We analyse the cases \u03c9=1,2,3 where the ratio\u2019s 1,2 turn out to be the most interesting. In an initial phase we find 2 adiabatic invariants with changes near the end of evolution. A remarkable feature is the vanishing and emergence of normal modes, stability changes and strong changes of the velocity distribution in phase-space. The problem is inspired by the dynamics of axisymmetric, rotating galaxies that evolve slowly to mirror symmetry with respect to the galactic plane, the model formulation is quite general.<\/jats:p>","DOI":"10.3390\/sym13071189","type":"journal-article","created":{"date-parts":[[2021,7,1]],"date-time":"2021-07-01T05:00:10Z","timestamp":1625115610000},"page":"1189","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Evolution to Mirror-Symmetry in Rotating Systems"],"prefix":"10.3390","volume":"13","author":[{"given":"Ferdinand","family":"Verhulst","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Utrecht, P.O. Box 80.010, 3508 TA Utrecht, The Netherlands"}]}],"member":"1968","published-online":{"date-parts":[[2021,7,1]]},"reference":[{"key":"ref_1","unstructured":"Roseau, M. (1966). 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Mech."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/7\/1189\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:24:35Z","timestamp":1760163875000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/7\/1189"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,7,1]]},"references-count":15,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2021,7]]}},"alternative-id":["sym13071189"],"URL":"https:\/\/doi.org\/10.3390\/sym13071189","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,7,1]]}}}