{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:44:54Z","timestamp":1760147094317,"version":"build-2065373602"},"reference-count":14,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2023,1,12]],"date-time":"2023-01-12T00:00:00Z","timestamp":1673481600000},"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 consider autonomous holonomic dynamical systems defined by equations of the form q\u00a8a=\u2212\u0393bca(q)q\u02d9bq\u02d9c\u2212Qa(q), where \u0393bca(q) are the coefficients of a symmetric (possibly non-metrical) connection and \u2212Qa(q) are the generalized forces. We prove a theorem which for these systems determines autonomous and time-dependent first integrals (FIs) of any order in a systematic way, using the \u2019symmetries\u2019 of the geometry defined by the dynamical equations. We demonstrate the application of the theorem to compute linear, quadratic, and cubic FIs of various Riemannian and non-Riemannian dynamical systems.<\/jats:p>","DOI":"10.3390\/sym15010222","type":"journal-article","created":{"date-parts":[[2023,1,13]],"date-time":"2023-01-13T02:05:52Z","timestamp":1673575552000},"page":"222","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Higher-Order First Integrals of Autonomous Non-Riemannian Dynamical Systems"],"prefix":"10.3390","volume":"15","author":[{"given":"Antonios","family":"Mitsopoulos","sequence":"first","affiliation":[{"name":"Faculty of Physics, Department of Astronomy-Astrophysics-Mechanics, University of Athens, Panepistemiopolis, 157 83 Athens, Greece"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7447-0046","authenticated-orcid":false,"given":"Michael","family":"Tsamparlis","sequence":"additional","affiliation":[{"name":"NITheCS, National Institute for Theoretical and Computational Sciences, Pietermaritzburg 3201, KwaZulu-Natal, South Africa"},{"name":"TCCMMP, Theoretical and Computational Condensed Matter and Materials Physics Group, School of Chemistry and Physics, University of KwaZulu-Natal, Pietermaritzburg 3201, KwaZulu-Natal, South Africa"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6712-7821","authenticated-orcid":false,"given":"Aniekan Magnus","family":"Ukpong","sequence":"additional","affiliation":[{"name":"NITheCS, National Institute for Theoretical and Computational Sciences, Pietermaritzburg 3201, KwaZulu-Natal, South Africa"},{"name":"TCCMMP, Theoretical and Computational Condensed Matter and Materials Physics Group, School of Chemistry and Physics, University of KwaZulu-Natal, Pietermaritzburg 3201, KwaZulu-Natal, South Africa"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,1,12]]},"reference":[{"doi-asserted-by":"crossref","unstructured":"Arnold, V.I. (1989). Mathematical Methods of Classical Mechanics, Springer.","key":"ref_1","DOI":"10.1007\/978-1-4757-2063-1"},{"unstructured":"Whittaker, E.T. (1917). A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, Cambridge University Press. [2nd ed.].","key":"ref_2"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1878","DOI":"10.1063\/1.525160","article-title":"Geodesic first integrals with explicit path-parameter dependence in Riemannian space-times","volume":"22","author":"Katzin","year":"1981","journal-title":"J. Math. Phys."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"3474","DOI":"10.1063\/1.526114","article-title":"Polynomial Constants of Motion in Flat Space","volume":"25","author":"Thompson","year":"1984","journal-title":"J. Math. Phys."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"102902","DOI":"10.1063\/1.2789555","article-title":"Higher order first integrals in classical mechanics","volume":"48","author":"Horwood","year":"2007","journal-title":"J. Math. Phys."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"405201","DOI":"10.1088\/1751-8113\/48\/40\/405201","article-title":"General Nth order integrals of motion in the Euclidean plane","volume":"48","author":"Post","year":"2015","journal-title":"J. Phys. A: Math. Gen."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"104383","DOI":"10.1016\/j.geomphys.2021.104383","article-title":"Higher order first integrals of autonomous dynamical systems","volume":"170","author":"Mitsopoulos","year":"2021","journal-title":"J. Geom. Phys."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1007\/BF01177666","article-title":"Noether\u2019s Theory in Classical Nonconservative Mechanics","volume":"23","author":"Djukic","year":"1975","journal-title":"Acta Mech."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"122701","DOI":"10.1063\/5.0029487","article-title":"First integrals of holonomic systems without Noether symmetries","volume":"61","author":"Tsamparlis","year":"2020","journal-title":"J. Math. Phys."},{"doi-asserted-by":"crossref","unstructured":"Mitsopoulos, A., and Tsamparlis, M. (2021). Quadratic first integrals of time-dependent dynamical systems of the form q\u00a8a=\u2212\u0393bcaq\u02d9bq\u02d9c\u2212\u03c9(t)Qa(q). Mathematics, 9.","key":"ref_10","DOI":"10.3390\/math9131503"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"104061","DOI":"10.1103\/PhysRevD.99.104061","article-title":"Integrability of geodesic motions in curved manifolds through nonlocal conserved charges","volume":"99","author":"Dimakis","year":"2019","journal-title":"Phys. Rev. D"},{"doi-asserted-by":"crossref","unstructured":"Mitsopoulos, A., and Tsamparlis, M. (2022). Quadratic first integrals of constrained autonomous conservative dynamical systems with fixed energy. Symmetry, 14.","key":"ref_12","DOI":"10.3390\/sym14091870"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"5666","DOI":"10.1103\/PhysRevA.41.5666","article-title":"Superintegrability in classical mechanics","volume":"41","author":"Evans","year":"1990","journal-title":"Phys. Rev. A"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1003","DOI":"10.1063\/1.1633352","article-title":"Hamiltonians separable in Cartesian coordinates and third-order integrals of motion","volume":"45","author":"Gravel","year":"2004","journal-title":"J. Math. Phys."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/1\/222\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T18:04:33Z","timestamp":1760119473000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/1\/222"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,1,12]]},"references-count":14,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2023,1]]}},"alternative-id":["sym15010222"],"URL":"https:\/\/doi.org\/10.3390\/sym15010222","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2023,1,12]]}}}