{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:15:28Z","timestamp":1760148928416,"version":"build-2065373602"},"reference-count":51,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2023,6,19]],"date-time":"2023-06-19T00:00:00Z","timestamp":1687132800000},"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 Lie system is a nonautonomous system of first-order ordinary differential equations whose general solution can be written via an autonomous function, the so-called (nonlinear) superposition rule of a finite number of particular solutions and some parameters to be related to initial conditions. This superposition rule can be obtained using the geometric features of the Lie system, its symmetries, and the symmetric properties of certain morphisms involved. Even if a superposition rule for a Lie system is known, the explicit analytic expression of its solutions frequently is not. This is why this article focuses on a novel geometric attempt to integrate Lie systems analytically and numerically. We focus on two families of methods based on Magnus expansions and on Runge\u2013Kutta\u2013Munthe\u2013Kaas methods, which are here adapted, in a geometric manner, to Lie systems. To illustrate the accuracy of our techniques we analyze Lie systems related to Lie groups of the form SL(n,R), which play a very relevant role in mechanics. In particular, we depict an optimal control problem for a vehicle with quadratic cost function. Particular numerical solutions of the studied examples are given.<\/jats:p>","DOI":"10.3390\/sym15061285","type":"journal-article","created":{"date-parts":[[2023,6,20]],"date-time":"2023-06-20T01:37:38Z","timestamp":1687225058000},"page":"1285","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Geometric Numerical Methods for Lie Systems and Their Application in Optimal Control"],"prefix":"10.3390","volume":"15","author":[{"given":"Luis","family":"Blanco D\u00edaz","sequence":"first","affiliation":[{"name":"Department of Applied Mathematics, Universidad Polit\u00e9cnica de Madrid (UPM), c. Jos\u00e9 Guti\u00e9rrez Abascal 2, 28006 Madrid, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Cristina","family":"Sard\u00f3n","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics, Universidad Polit\u00e9cnica de Madrid (UPM), c. Jos\u00e9 Guti\u00e9rrez Abascal 2, 28006 Madrid, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Fernando","family":"Jim\u00e9nez Alburquerque","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics, Universidad Polit\u00e9cnica de Madrid (UPM), c. Jos\u00e9 Guti\u00e9rrez Abascal 2, 28006 Madrid, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8643-144X","authenticated-orcid":false,"given":"Javier","family":"de Lucas","sequence":"additional","affiliation":[{"name":"Department of Mathematical Methods in Physics, University of Warsaw, ul. Pasteura 5, 02-093 Warsaw, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,6,19]]},"reference":[{"key":"ref_1","unstructured":"Cari\u00f1ena, J.F., Grabowski, J., and Marmo, G. (2000). Lie-Scheffers Systems: A Geometric Approach, Bibliopolis."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"de Lucas, J., and Sard\u00f3n, C. (2020). A Guide to Lie Systems with Compatible Geometric Structures, World Scientific.","DOI":"10.1142\/q0208"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"105","DOI":"10.1016\/0378-4371(82)90268-0","article-title":"Nonlinear action of Lie groups and superposition rules for nonlinear differential equations","volume":"114","author":"Winternitz","year":"1982","journal-title":"Phys. A"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1023\/A:1010743114995","article-title":"Reduction of t-dependent systems admitting a superposition principle","volume":"66","author":"Grabowski","year":"2001","journal-title":"Acta Appl. Math."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Cari\u00f1ena, J.F., and de Lucas, J. (2011). Lie Systems: Theory, Generalisations, and Applications. Diss. Math., 479.","DOI":"10.4064\/dm479-0-1"},{"key":"ref_6","unstructured":"Sard\u00f3n, C. (2015). Lie Systems, Lie Symmetries and Reciprocal Transformations. [Ph.D. Thesis, Universidad de Salamanca]."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1365","DOI":"10.1088\/0951-7715\/14\/5\/322","article-title":"Non-holonomic integrators","volume":"14","year":"2001","journal-title":"Nonlinearity"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1017\/S0962492900002154","article-title":"Lie-group methods","volume":"9","author":"Iserles","year":"2005","journal-title":"Acta Numer."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1313","DOI":"10.1088\/0951-7715\/19\/6\/006","article-title":"Discrete Lagrangian and Hamiltonian mechanics on Lie groupoids","volume":"19","author":"Marrero","year":"2006","journal-title":"Nonlinearity"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"357","DOI":"10.1017\/S096249290100006X","article-title":"Discrete mechanics and variational integrators","volume":"10","author":"Marsden","year":"2001","journal-title":"Acta Numer."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1017\/S0962492902000053","article-title":"Splitting methods","volume":"11","author":"McLachlan","year":"2002","journal-title":"Acta Numer."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Sanz-Serna, J.M. (1992). Symplectic integrators for Hamiltonian problems: An overview. Acta Numer., 243\u2013286.","DOI":"10.1017\/S0962492900002282"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"551","DOI":"10.1007\/s10883-012-9159-y","article-title":"Explicit solutions of the a1-type Lie-Scheffers system and a general Riccati equation","volume":"18","author":"Pietrzkowski","year":"2012","journal-title":"J. Dyn. Control Syst."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"305","DOI":"10.1016\/0010-4655(84)90136-X","article-title":"Nonlinear superposition principles: A new numerical method for solving matrix Riccati equations","volume":"33","author":"Rand","year":"1984","journal-title":"Comput. Phys. Commun."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1016\/S0034-4877(07)80137-6","article-title":"Superposition rules, Lie theorem and partial differential equations","volume":"60","author":"Grabowski","year":"2007","journal-title":"Rep. Math. Phys."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1935","DOI":"10.1142\/S0217751X9900097X","article-title":"Integrability of the Riccati equation from a group theoretical viewpoint","volume":"14","author":"Ramos","year":"1999","journal-title":"Int. J. Mod. Phys. A"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"034105","DOI":"10.1103\/PhysRevA.72.034105","article-title":"Two-level quantum dynamics, integrability and unitary NOT gates","volume":"72","author":"Angelo","year":"2005","journal-title":"Phys. Rev. A"},{"key":"ref_18","first-page":"910","article-title":"Superposition rules and stochastic Lie-Scheffers systems","volume":"45","author":"Ortega","year":"2009","journal-title":"Ann. Inst. H. Poincar\u00e9 Probab. Stat."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"2528","DOI":"10.1063\/1.528997","article-title":"Superposition formulas for nonlinear superequations","volume":"31","author":"Hussin","year":"1990","journal-title":"J. Math. Phys."},{"key":"ref_20","first-page":"1260007","article-title":"A new Lie systems approach to second-order Riccati equations","volume":"9","year":"2011","journal-title":"Int. J. Geom. Meth. Mod. Phys."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"683","DOI":"10.1142\/S0219887809003758","article-title":"Applications of Lie systems in dissipative Milne-Pinney equations","volume":"6","year":"2009","journal-title":"Int. J. Geom. Meth. Mod. Phys."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"293","DOI":"10.1016\/S0034-4877(00)89038-2","article-title":"The Superposition Principle for the Lie Type first-order PDEs","volume":"45","author":"Odzijewicz","year":"2000","journal-title":"Rep. Math. Phys."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"983","DOI":"10.1098\/rsta.1999.0362","article-title":"On the solution of linear differential equations in Lie groups","volume":"357","author":"Iserles","year":"1999","journal-title":"Philos. Trans. R. Soc. A"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1145","DOI":"10.1137\/S0036142997326616","article-title":"Collocation and relaxed collocation for the Fer and Magnus expansions","volume":"36","author":"Zanna","year":"1999","journal-title":"J. Numer. Anal."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"92","DOI":"10.1007\/BF02510919","article-title":"Runge-Kutta methods on Lie groups","volume":"38","year":"1998","journal-title":"BIT Numer. Math."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1016\/S0168-9274(98)00030-0","article-title":"High order Runge-Kutta methods on manifolds","volume":"29","year":"1999","journal-title":"J. Appl. Numer. Math."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"299","DOI":"10.1016\/j.jde.2017.02.038","article-title":"A Lie systems approach to the Riccati hierarchy and partial differential equations","volume":"263","author":"Grundland","year":"2017","journal-title":"J. Differ. Equ."},{"key":"ref_28","first-page":"42","article-title":"A Review of the Matrix Riccati Equation","volume":"9","year":"1973","journal-title":"Kybernetika"},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Lee, J.M. (2003). Introduction to Smooth Manifolds, Springer. Graduate Texts in Mathematics 218.","DOI":"10.1007\/978-0-387-21752-9"},{"key":"ref_30","first-page":"159","article-title":"The representation of Lie algebras by matrices","volume":"2","author":"Ado","year":"1947","journal-title":"Uspekhi Mat. Nauk."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Curtis, M.L. (1984). Matrix Groups, Springer. [2nd ed.].","DOI":"10.1007\/978-1-4612-5286-3"},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Hall, B. (2015). Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Springer International Publishing.","DOI":"10.1007\/978-3-319-13467-3"},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Sattinger, D.H., and Weaver, O.L. (1986). Lie Groups and Algebras with Applications to Physics, Springer. Geometry and Mechanics.","DOI":"10.1007\/978-1-4757-1910-9"},{"key":"ref_34","unstructured":"Lie, S., and Scheffers, G. (1893). Vorlesungen \u00fcber continuierliche Gruppen mit geometrischen und anderen Anwendungen, Teubner."},{"key":"ref_35","first-page":"551","article-title":"Sulla Struttura dei Gruppi Finiti e Continui","volume":"40","author":"Levi","year":"1905","journal-title":"Atti Della R. Accad. Delle Sci. Torino"},{"key":"ref_36","unstructured":"Hairer, E., N\u00f8rsett, S.P., and Wanner, G. (1993). Solving Ordinary Differential Equations I: Nonstiff Problems, Springer."},{"key":"ref_37","unstructured":"Isaacson, E., and Keller, H.B. (1966). Analysis of Numerical Methods, John Wiley & Sons."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Quarteroni, A., Sacco, R., and Saleri, F. (2007). Numerical Mathematics, Springer.","DOI":"10.1007\/978-0-387-22750-4"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"649","DOI":"10.1002\/cpa.3160070404","article-title":"On the exponential solution of differential equations for a linear operator","volume":"7","author":"Magnus","year":"1954","journal-title":"Commun. Pure Appl. Math."},{"key":"ref_40","unstructured":"Iserles, A., N\u00f8rsett, S.P., and Rasmussen, A.F. (1998). t-Symmetry and High-Order Magnus Methods, University of Cambridge. Technical Report 1998\/NA06, DAMTP."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"434","DOI":"10.1023\/A:1022311628317","article-title":"Improved high order integrators based on the Magnus expansion","volume":"40","author":"Blanes","year":"2000","journal-title":"BIT Numer. Math."},{"key":"ref_42","unstructured":"Hairer, E., Lubich, C., and Wanner, G. (2006). Geometric Numerical Integration, Springer."},{"key":"ref_43","unstructured":"Hartshorne, R. (1967). Foundations of Projective Geometry, W.A. Benjamin, Inc."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"1062","DOI":"10.1063\/1.525831","article-title":"Superposition principles for matrix Riccati equations","volume":"24","author":"Harnad","year":"1983","journal-title":"J. Math. Phys."},{"key":"ref_45","unstructured":"Reid, W.T. (1972). Riccati Differential Equations, Academic."},{"key":"ref_46","unstructured":"Dom\u00ednguez, S., Campoy, P., Sebasti\u00e1n, J.M., and Jim\u00e9nez, A. (2006). Control en el Espacio de Estado, Pearson."},{"key":"ref_47","unstructured":"Sontag, E.D. (1998). Mathematical Control Theory: Deterministic Finite Dimensional Systems, Springer."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1016\/j.ijnonlinmec.2017.04.004","article-title":"Chiellini integrability and quadratically damped oscillators","volume":"92","author":"Pandey","year":"2017","journal-title":"Int. J. Non-Linear Mech."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"533","DOI":"10.1016\/j.jmaa.2004.02.028","article-title":"Discrete matrix Riccati equations with super-position formulas","volume":"294","author":"Penskoi","year":"2004","journal-title":"J. Math. Anal. Appl."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"495201","DOI":"10.1088\/1751-8121\/aa918f","article-title":"Lie-Hamilton systems on curved spaces: A geometrical approach","volume":"50","author":"Herranz","year":"2017","journal-title":"J. Phys. A"},{"key":"ref_51","doi-asserted-by":"crossref","unstructured":"Lange, J., and de Lucas, J. (2019). Geometric models for Lie\u2013Hamilton systems on R2. Mathematics, 7.","DOI":"10.3390\/math7111053"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/6\/1285\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T19:56:45Z","timestamp":1760126205000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/6\/1285"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,6,19]]},"references-count":51,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2023,6]]}},"alternative-id":["sym15061285"],"URL":"https:\/\/doi.org\/10.3390\/sym15061285","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2023,6,19]]}}}