{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T01:08:15Z","timestamp":1760231295528,"version":"build-2065373602"},"reference-count":40,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2022,9,7]],"date-time":"2022-09-07T00:00:00Z","timestamp":1662508800000},"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 conservative dynamical systems which are constrained with the condition that the total energy of the system has a specified value. We prove a theorem which provides the quadratic first integrals (QFIs), time-dependent and autonomous, of these systems in terms of the symmetries (conformal Killing vectors and conformal Killing tensors) of the kinetic metric. It is proved that there are three types of QFIs and for each type we give explicit formulae for their computation. It is also shown that when the autonomous QFIs are considered, then we recover the known results of previous works. For a zero potential function, we have the case of constrained geodesics and obtain formulae to compute their QFIs. The theorem is applied in two cases. In the first case, we determine potentials which admit the second of the three types of QFIs. We recover a superintegrable potential of the Ermakov type and a new integrable potential whose trajectories for zero energy and zero QFI are circles. In the second case, we integrate the constrained geodesic equations for a family of two-dimensional conformally flat metrics.<\/jats:p>","DOI":"10.3390\/sym14091870","type":"journal-article","created":{"date-parts":[[2022,9,8]],"date-time":"2022-09-08T09:51:09Z","timestamp":1662630669000},"page":"1870","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Quadratic First Integrals of Constrained Autonomous Conservative Dynamical Systems with Fixed Energy"],"prefix":"10.3390","volume":"14","author":[{"given":"Antonios","family":"Mitsopoulos","sequence":"first","affiliation":[{"name":"Department of Astronomy-Astrophysics-Mechanics, Faculty of Physics, University of Athens, Panepistemiopolis, 15783 Athens, Greece"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"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"}]}],"member":"1968","published-online":{"date-parts":[[2022,9,7]]},"reference":[{"key":"ref_1","first-page":"273","article-title":"A third integral of motion in a galaxy","volume":"49","author":"Contopoulos","year":"1960","journal-title":"Z. Astrophys."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1093\/mnras\/124.1.1","article-title":"Stellar Dynamics: Only isolating integrals should be used in Jeans\u2019 Theorem","volume":"124","year":"1962","journal-title":"Mon. Not. R. Astron. Soc."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1086\/108903","article-title":"On the Existence of a Third Integral of Motion","volume":"68","author":"Contopoulos","year":"1963","journal-title":"Astron. J."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1086\/109234","article-title":"The Applicability of the Third Integral of Motion: Some Numerical Experiments","volume":"69","author":"Heiles","year":"1964","journal-title":"Astron. J."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"472","DOI":"10.3934\/mine.2020022","article-title":"A Review of the \u201cThird\u201d Integral","volume":"2","author":"Contopoulos","year":"2020","journal-title":"Math. Eng."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"591","DOI":"10.2307\/1968307","article-title":"Dynamical trajectories and geodesics","volume":"30","author":"Eisenhart","year":"1928","journal-title":"Ann. Math."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Benn, I.M. (2006). Geodesics and Killing tensors in mechanics. J. Math. Phys., 47.","DOI":"10.1063\/1.2168121"},{"key":"ref_8","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_9","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_10","doi-asserted-by":"crossref","first-page":"269","DOI":"10.1016\/0001-8708(75)90139-5","article-title":"Curvature and Mechanics","volume":"15","author":"Pin","year":"1975","journal-title":"Adv. Math."},{"key":"ref_11","unstructured":"Abraham, R., and Marsden, J.E. (1978). Foundations of Mechanics, Addison-Wesley Publishing Company, Inc."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"3235","DOI":"10.1088\/0305-4470\/28\/11\/021","article-title":"Invariants at fixed and arbitrary energy. A unified geometric approach","volume":"28","author":"Rosquist","year":"1995","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"370","DOI":"10.1063\/1.533137","article-title":"A unified treatment of cubic invariants at fixed and arbitrary energy","volume":"41","author":"Karlovini","year":"2000","journal-title":"J. Math. Phys."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"4041","DOI":"10.1063\/1.1483107","article-title":"A unified treatment of quartic invariants at fixed and arbitrary energy","volume":"43","author":"Karlovini","year":"2002","journal-title":"J. Math. Phys."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"052902","DOI":"10.1063\/1.1888565","article-title":"Configurational invariants of Hamiltonian systems","volume":"46","author":"Pucacco","year":"2005","journal-title":"J. Math. Phys."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"8","DOI":"10.1063\/1.1664480","article-title":"Related First Integral Theorem: A Method for Obtaining Conservation Laws of Dynamical Systems with Geodesic Trajectories in Riemannian Spaces Admitting Symmetries","volume":"9","author":"Katzin","year":"1968","journal-title":"J. Math. Phys."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1213","DOI":"10.1063\/1.1666467","article-title":"Related integral theorem II. A method for obtaining quadratic constants of the motion for conservative dynamical systems admitting symmetries","volume":"14","author":"Katzin","year":"1973","journal-title":"J. Math. Phys."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1886","DOI":"10.1063\/1.1666264","article-title":"Symmetry mappings of constrained dynamical systems and an associated realted integral theorem","volume":"14","author":"Levine","year":"1973","journal-title":"J. Math. Phys."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"175202","DOI":"10.1088\/1751-8113\/44\/17\/175202","article-title":"Two-dimensional dynamical systems which admit Lie and Noether symmetries","volume":"44","author":"Tsamparlis","year":"2011","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Hodge, W.V.D., and Pedoe, D. (1994). Methods of Algebraic Geometry, Cambridge University Press.","DOI":"10.1017\/CBO9780511623899"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"012021","DOI":"10.1088\/1742-6596\/670\/1\/012021","article-title":"Contact symmetries of constrained quadratic Lagrangians","volume":"670","author":"Dimakis","year":"2016","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_22","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"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"024043","DOI":"10.1103\/PhysRevD.106.024043","article-title":"Hidden symmetries from distortions of the conformal structure","volume":"106","author":"Dimakis","year":"2022","journal-title":"Phys. Rev. D"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"129","DOI":"10.4153\/CJM-1950-012-1","article-title":"Generalized Hamiltonian Dynamics","volume":"2","author":"Dirac","year":"1950","journal-title":"Can. J. Math."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1018","DOI":"10.1103\/PhysRev.83.1018","article-title":"Constraints in Covariant Field Theories","volume":"83","author":"Anderson","year":"1951","journal-title":"Phys. Rev."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"326","DOI":"10.1098\/rspa.1958.0141","article-title":"Generalized Hamiltonian Dynamics","volume":"246","author":"Dirac","year":"1958","journal-title":"Proc. R. Soc. Lond. A"},{"key":"ref_27","unstructured":"Dirac, P.A.M. (1964). Lectrures on Quantum Mechanics, Yeshiva University Press."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"072703","DOI":"10.1063\/1.5141392","article-title":"Quadratic first integrals of autonomous conservative dynamical systems","volume":"61","author":"Tsamparlis","year":"2020","journal-title":"J. Math. Phys."},{"key":"ref_29","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."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"52","DOI":"10.1016\/j.geomphys.2015.12.003","article-title":"Variational contact symmetries of constrained Lagrangians","volume":"101","author":"Terzis","year":"2016","journal-title":"J. Geom. Phys."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1929","DOI":"10.1088\/0264-9381\/20\/11\/301","article-title":"Killing tensors and conformal Killing tensors from conformal Killing vectors","volume":"20","author":"Rani","year":"2003","journal-title":"Class. Quantum Gravity"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"133","DOI":"10.1016\/0375-9601(83)90195-0","article-title":"Homothetic Killing tensors","volume":"97","author":"Prince","year":"1983","journal-title":"Phys. Lett. A"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"265","DOI":"10.1007\/BF01649445","article-title":"On Quadratic First Integrals of the Geodesic Equations for Type {22} Spacetimes","volume":"18","author":"Walker","year":"1970","journal-title":"Commun. Math. Phys."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1782","DOI":"10.1063\/1.523488","article-title":"Conformal Killing tensors in reducible spaces","volume":"18","author":"Weir","year":"1977","journal-title":"J. Math. Phys."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1016\/0370-1573(87)90089-5","article-title":"Direct methods for the search of the second invariant","volume":"147","author":"Hietarinta","year":"1987","journal-title":"Phys. Rep."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Mitsopoulos, A., Tsamparlis, M., and Paliathanasis, A. (2020). Integrable and superintegrable potentials of 2d autonomous conservative dynamical systems. Symmetry, 12.","DOI":"10.3390\/sym12101655"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1023\/A:1008240112483","article-title":"Lie-B\u00e4cklund and Noether Symmetries with Applications","volume":"15","author":"Ibragimov","year":"1998","journal-title":"Nonlinear Dyn."},{"key":"ref_38","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\u0393abcq\u02d9bq\u02d9c\u2212\u03c9(t)Qa(q). Mathematics, 9.","DOI":"10.3390\/math9131503"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"335","DOI":"10.1007\/BF02575448","article-title":"Integration of D-dimensional cosmological models with two factor spaces by reduction to the generalized Emden-Fowler equation","volume":"114","author":"Gavrilov","year":"1998","journal-title":"Theor. Math. Phys."},{"key":"ref_40","first-page":"371","article-title":"Sur un probl\u00e9me de m\u00e8canique","volume":"6","author":"Darboux","year":"1901","journal-title":"Arch. Neerl. Sci."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/9\/1870\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:25:01Z","timestamp":1760142301000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/9\/1870"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,9,7]]},"references-count":40,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2022,9]]}},"alternative-id":["sym14091870"],"URL":"https:\/\/doi.org\/10.3390\/sym14091870","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2022,9,7]]}}}