# American Institute of Mathematical Sciences

January  2011, 29(1): 1-24. doi: 10.3934/dcds.2011.29.1

## Non-integrability of the degenerate cases of the Swinging Atwood's Machine using higher order variational equations

 1 Departament de Matemàtiques, Universitat Autònoma de Barcelona, Bellaterra, Barcelona, Spain 2 Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, Gran Via 585, 08007 Barcelona

Received  January 2010 Revised  July 2010 Published  September 2010

Non-integrability criteria, based on differential Galois theory and requiring the use of higher order variational equations (VEk), are applied to prove the non-integrability of the Swinging Atwood's Machine for values of the parameter which can not be decided using first order variational equations (VE1).
Citation: Regina Martínez, Carles Simó. Non-integrability of the degenerate cases of the Swinging Atwood's Machine using higher order variational equations. Discrete and Continuous Dynamical Systems, 2011, 29 (1) : 1-24. doi: 10.3934/dcds.2011.29.1
##### References:
 [1] M. Abramowitz and I. A. Stegun, "Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables," Reprint of the 1972 edition. Dover Publications, Inc., New York, 1992. [2] C. Batut, K. Belabas, D. Bernardi, H. Cohen and M. Olivier, Users' guide to PARI/GP, (freely available from http://pari.math.u-bordeaux.fr/). [3] J. Casasayas, A. Nunes and N. Tufillaro, Swinging Atwood's machine, J. Physique, 51 (1990), 1693-1702. [4] J. Martinet and J-P. Ramis, Théorie de Galois différentielle et resommation, "Computer Algebra and Differential Equations," 117-214, E.Tournier, Ed., Academic Press, London, 1989. [5] R. Martínez and C. Simó, Non-integrability of Hamiltonian systems through high order variational equations: Summary of results and examples, Regular and Chaotic Dynamics, 14 (2009), 323-348. [6] R. Martínez and C. Simó, Efficient numerical implementation of integrability criteria based on high order variational equations, Preprint, 2010. [7] J. J. Morales-Ruiz, "Differential Galois Theory and Non-Integrability of Hamiltonian Systems," Progress in Mathematics vol. 179, Birkhäuser, 1999. [8] J. J. Morales-Ruiz and J.-P. Ramis, Galoisian obstructions to integrability of Hamiltonian systems, Methods and Applications of Analysis, 8 (2001), 33-96. [9] J. J. Morales-Ruiz and J.-P. Ramis, Galoisian Obstructions to integrability of Hamiltonian systems II, Methods and Applications of Analysis, 8 (2001), 97-112. [10] J. J. Morales-Ruiz, J.-P. Ramis and C. Simó, Integrability of Hamiltonian systems and differential Galois groups of higher variational equations, Ann. Scient. Éc. Norm. Sup. 4$^e$ série, 40 (2007), 845-884. [11] J. J. Morales and C. Simó, Non integrability criteria for Hamiltonians in the case of Lamé normal variational equations, J. Diff. Equations, 129 (1996), 111-135. doi: doi:10.1006/jdeq.1996.0113. [12] J. J. Morales, C. Simó and S. Simón, Algebraic proof of the non-integrability of Hill's Problem, Ergodic Theory and Dynamical Systems, 25 (2005), 1237-1256. doi: doi:10.1017/S0143385704001038. [13] O. Pujol, J.-P. Pérez, J.-P. Ramis, C. Simó, S. Simon and J.-A. Weil, Swinging Atwood's Machine: Experimental and numerical results, theoretical study, Physica D, 239 (2010), 1067-1081. doi: doi:10.1016/j.physd.2010.02.017. [14] L. Schlesinger, "Handbuch der Theorie der Linearen Differentialgleichungen," Reprint. Biblioteca Mathematica Teubneriana, Band 31, Johnson Reprint Corp., New York - London, 1968. [15] N. Tufillaro, Integrable motion of a swinging Atwood's machine, Am. J. Phys., 54 (1986), 142-143. doi: doi:10.1119/1.14710. [16] S. L. Ziglin, Branching of solutions and non-existence of first integrals in Hamiltonian mechanics I, Funct. Anal. Appl., 16 (1982), 181-189. doi: doi:10.1007/BF01081586.

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##### References:
 [1] M. Abramowitz and I. A. Stegun, "Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables," Reprint of the 1972 edition. Dover Publications, Inc., New York, 1992. [2] C. Batut, K. Belabas, D. Bernardi, H. Cohen and M. Olivier, Users' guide to PARI/GP, (freely available from http://pari.math.u-bordeaux.fr/). [3] J. Casasayas, A. Nunes and N. Tufillaro, Swinging Atwood's machine, J. Physique, 51 (1990), 1693-1702. [4] J. Martinet and J-P. Ramis, Théorie de Galois différentielle et resommation, "Computer Algebra and Differential Equations," 117-214, E.Tournier, Ed., Academic Press, London, 1989. [5] R. Martínez and C. Simó, Non-integrability of Hamiltonian systems through high order variational equations: Summary of results and examples, Regular and Chaotic Dynamics, 14 (2009), 323-348. [6] R. Martínez and C. Simó, Efficient numerical implementation of integrability criteria based on high order variational equations, Preprint, 2010. [7] J. J. Morales-Ruiz, "Differential Galois Theory and Non-Integrability of Hamiltonian Systems," Progress in Mathematics vol. 179, Birkhäuser, 1999. [8] J. J. Morales-Ruiz and J.-P. Ramis, Galoisian obstructions to integrability of Hamiltonian systems, Methods and Applications of Analysis, 8 (2001), 33-96. [9] J. J. Morales-Ruiz and J.-P. Ramis, Galoisian Obstructions to integrability of Hamiltonian systems II, Methods and Applications of Analysis, 8 (2001), 97-112. [10] J. J. Morales-Ruiz, J.-P. Ramis and C. Simó, Integrability of Hamiltonian systems and differential Galois groups of higher variational equations, Ann. Scient. Éc. Norm. Sup. 4$^e$ série, 40 (2007), 845-884. [11] J. J. Morales and C. Simó, Non integrability criteria for Hamiltonians in the case of Lamé normal variational equations, J. Diff. Equations, 129 (1996), 111-135. doi: doi:10.1006/jdeq.1996.0113. [12] J. J. Morales, C. Simó and S. Simón, Algebraic proof of the non-integrability of Hill's Problem, Ergodic Theory and Dynamical Systems, 25 (2005), 1237-1256. doi: doi:10.1017/S0143385704001038. [13] O. Pujol, J.-P. Pérez, J.-P. Ramis, C. Simó, S. Simon and J.-A. Weil, Swinging Atwood's Machine: Experimental and numerical results, theoretical study, Physica D, 239 (2010), 1067-1081. doi: doi:10.1016/j.physd.2010.02.017. [14] L. Schlesinger, "Handbuch der Theorie der Linearen Differentialgleichungen," Reprint. Biblioteca Mathematica Teubneriana, Band 31, Johnson Reprint Corp., New York - London, 1968. [15] N. Tufillaro, Integrable motion of a swinging Atwood's machine, Am. J. Phys., 54 (1986), 142-143. doi: doi:10.1119/1.14710. [16] S. L. Ziglin, Branching of solutions and non-existence of first integrals in Hamiltonian mechanics I, Funct. Anal. Appl., 16 (1982), 181-189. doi: doi:10.1007/BF01081586.
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