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

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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

Regina Martínez ^1, and Carles Simó ^2,

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

Abstract
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Non-integrability criteria, based on differential Galois theory and requiring the use of higher order variational equations (VE_k), 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).

Keywords: differential Galois theory, Non-integrability criteria, Swinging Atwood Machine, higher order variationals..

Mathematics Subject Classification: Primary: 37J30, 70H07; Secondary: 34M3.

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.

show all references

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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