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.

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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