{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T13:29:39Z","timestamp":1787232579243,"version":"build-2736575974"},"reference-count":38,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Appl. Dyn. Syst."],"published-print":{"date-parts":[[2009,1]]},"abstract":"<jats:p>In this paper we present an approach toward the comprehensive analysis of the nonintegrability of differential equations in the form $\\ddot{x}=f(x,t)$ which is analogous to Hamiltonian systems with $1+1\/2$ degrees of freedom. In particular, we analyze the nonintegrability of some important families of differential equations such as Painlev\u00e9 II, Sitnikov, and the Hill\u2013Schr\u00f6dinger equation. We emphasize Painlev\u00e9 II, showing its nonintegrability through three different Hamiltonian systems, and also Sitnikov, in which two different versions including numerical results are shown. The main tool for studying the nonintegrability of these kinds of Hamiltonian systems is Morales\u2013Ramis theory. This paper is a very slight improvement to the talk with a similar title delivered by the author at the SIAM Conference on Applications of Dynamical Systems in 2007.<\/jats:p>","DOI":"10.1137\/080730329","type":"journal-article","created":{"date-parts":[[2009,2,13]],"date-time":"2009-02-13T18:17:12Z","timestamp":1234549032000},"page":"279-297","source":"Crossref","is-referenced-by-count":10,"title":["Nonautonomous Hamiltonian Systems and Morales\u2013Ramis Theory I. The Case $\\ddot{x}=f(x,t)$"],"prefix":"10.1137","volume":"8","author":[{"given":"Primitivo B.","family":"Acosta-Hum\u00e1nez","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2009,2,13]]},"reference":[{"key":"R1","unstructured":"R. Abraham and J. E. Marsden,\n                      Foundations of Mechanics\n                      , 2nd ed., revised and enlarged, Benjamin\/ Cummings, Advanced Book Program, Reading, MA, 1978."},{"key":"R2","unstructured":"M. Abramowitz and I. A. Stegun, eds.\n                      Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables\n                      , John Wiley & Sons, New York, 1984. Reprint of the 1972 edition."},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.3934\/dcdsb.2008.10.265"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1023\/A:1019979414586"},{"key":"R5","doi-asserted-by":"crossref","unstructured":"V. I. Arnold,\n                      Mathematical Methods of Classical Mechanics\n                      , Graduate Texts in Mathematics 60, Springer-Verlag, New York, 1978.","DOI":"10.1007\/978-1-4757-1693-1"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.14.2338"},{"key":"R7","unstructured":"D. Boucher,\n                      Sur les \u00e9quations diff\u00e9rentielles lin\u00e9aires param\u00e9tr\u00e9es; une application aux syst\u00e8mes hamiltoniens\n                      , Th\u00e8se de l'Universit\u00e9 de Limoges, Limoges, France, 2000."},{"key":"R8","doi-asserted-by":"crossref","unstructured":"D. Boucher and J.A. Weil,\n                      Application of J.J. Morales and J.P. Ramis' theorem to test the non-complete integrability of the planar three-body problem\n                      , in From Combinatorics to Dynamical Systems, IRMA Lect. Math. Theor. Phys. 3, de Gruyter, Berlin, 2003, pp. 163\u2013177.","DOI":"10.1515\/9783110200003.163"},{"key":"R9","doi-asserted-by":"crossref","unstructured":"A. Dewisme, S. Bouquet, and P. G. L. Leach,\n                      Symmetries of time dependent Hamiltonian systems\n                      , in Modern Group Analysis: Advanced Analytical and Computational Methods in Mathematical Physics (Acireale, 1992), Kluwer, Dordrecht, The Netherlands, 1993, pp. 173\u2013179.","DOI":"10.1007\/978-94-011-2050-0_16"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1016\/j.crma.2006.06.006"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1063\/1.530206"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1088\/0143-0807\/17\/5\/009"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.3934\/dcds.2004.11.843"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1088\/0305-4470\/35\/29\/102"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1016\/j.cnsns.2004.07.002"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1007\/BF00699722"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1134\/S1560354707060056"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1023\/A:1026605011434"},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1016\/0034-4877(84)90023-5"},{"key":"R20","unstructured":"C. Lanczos,\n                      The Variational Principles of Mechanics\n                      , Mathematical Expositions 4, University of Toronto Press, Toronto, Ontario, Canada, 1949."},{"key":"R21","doi-asserted-by":"crossref","unstructured":"A. J. Lichtenberg and M. A. Lieberman,\n                      Regular and Chaotic Dynamics\n                      , 2nd ed., Applied Mathematical Sciences 38, Springer-Verlag, New York, 1992.","DOI":"10.1007\/978-1-4757-2184-3"},{"key":"R22","doi-asserted-by":"crossref","unstructured":"J. J. Morales-Ruiz,\n                      Differential Galois Theory and Non-Integrability of Hamiltonian Systems\n                      , Progress in Mathematics 179, Birkh\u00e4user Verlag, Basel, 1999.","DOI":"10.1007\/978-3-0348-0723-4"},{"key":"R23","unstructured":"J. J. Morales-Ruiz,\n                      A remark about the Painlev\u00e9 transcendents\n                      , in Th\u00e9ories asymptotiques et \u00e9quations de Painlev\u00e9, S\u00e9min. Congr. 14, Soc. Math. France, Paris, 2006, pp. 229\u2013235."},{"key":"R24","doi-asserted-by":"publisher","DOI":"10.5802\/afst.925"},{"key":"R25","doi-asserted-by":"crossref","first-page":"33","DOI":"10.4310\/MAA.2001.v8.n1.a3","volume":"8","author":"Morales-Ruiz J. J.","year":"2001","journal-title":"Methods Appl. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/1073-2772","issn-type":"print"},{"key":"R26","doi-asserted-by":"crossref","first-page":"97","DOI":"10.4310\/MAA.2001.v8.n1.a4","volume":"8","author":"Morales-Ruiz J. J.","year":"2001","journal-title":"Methods Appl. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/1073-2772","issn-type":"print"},{"key":"R27","doi-asserted-by":"crossref","unstructured":"J. Moser,\n                      Stable and Random Motions in Dynamical Systems. With Special Emphasis on Celestial Mechanics\n                      , Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 2001. Reprint of the 1973 original.","DOI":"10.1515\/9781400882694"},{"key":"R28","doi-asserted-by":"crossref","unstructured":"E. L. Stiefel and G. Scheifele,\n                      Linear and Regular Celestial Mechanics. Perturbed Two-Body Motion, Numerical Methods, Canonical Theory\n                      , Grundlehren Math. Wiss. 174, Springer-Verlag, New York, 1971.","DOI":"10.1007\/978-3-642-65027-7_9"},{"key":"R29","first-page":"13","volume":"60","author":"Stoyanova T.","year":"2007","journal-title":"C. R. Acad. Bulgare Sci."},{"key":"R30","doi-asserted-by":"publisher","DOI":"10.1088\/0305-4470\/38\/6\/006"},{"key":"R31","doi-asserted-by":"publisher","DOI":"10.1002\/1521-3889(200201)11:1<15::AID-ANDP15>3.0.CO;2-0"},{"key":"R32","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.66.066605"},{"key":"R33","doi-asserted-by":"publisher","DOI":"10.1063\/1.532458"},{"key":"R34","doi-asserted-by":"crossref","unstructured":"W. Thirring,\n                      Lehrbuch der mathematischen Physik. Band\n                      1.\n                      Klassische dynamische Systeme\n                      , Springer-Verlag, Vienna, 1977.","DOI":"10.1007\/978-3-7091-3405-4_1"},{"key":"R35","doi-asserted-by":"publisher","DOI":"10.1088\/0305-4470\/33\/22\/318"},{"key":"R36","doi-asserted-by":"crossref","unstructured":"M. van der Put and M. F. Singer,\n                      Galois Theory of Linear Differential Equations\n                      , Grundlehren Math. Wiss. 328, Springer-Verlag, Berlin, 2003.","DOI":"10.1007\/978-3-642-55750-7"},{"key":"R37","unstructured":"E. T. Whittaker,\n                      A Treatise on the Analytical Dynamics of Particles and Rigid Bodies. With an Introduction to the Problem of Three Bodies\n                      , Cambridge Mathematical Library, Cambridge University Press, Cambridge, UK, 1988. Reprint of the 1937 edition."},{"key":"R38","doi-asserted-by":"crossref","unstructured":"K. Wodnar,\n                      New formulations of the Sitnikov problem\n                      , in Predictability, Stability, and Chaos in\n                      N\n                      -Body Dynamical Systems (Cortina d'Ampezzo, 1990), NATO Adv. Sci. Inst. Ser. B Phys. 272, Plenum, New York, 1991, pp. 457\u2013466.","DOI":"10.1007\/978-1-4684-5997-5_39"}],"container-title":["SIAM Journal on Applied Dynamical Systems"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/080730329","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T13:04:55Z","timestamp":1787231095000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/080730329"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,1]]},"references-count":38,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2009,1]]}},"alternative-id":["10.1137\/080730329"],"URL":"https:\/\/doi.org\/10.1137\/080730329","relation":{},"ISSN":["1536-0040"],"issn-type":[{"value":"1536-0040","type":"electronic"}],"subject":[],"published":{"date-parts":[[2009,1]]}}}