Non-integrability criterium for normal variational equations around an integrable subsystem and an example: The Wilberforce spring-pendulum

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March 2013, 33(3): 965-986. doi: 10.3934/dcds.2013.33.965

Non-integrability criterium for normal variational equations around an integrable subsystem and an example: The Wilberforce spring-pendulum

Primitivo B. Acosta-Humánez ^1, , Martha Alvarez-Ramírez ^2, , David Blázquez-Sanz ^3, and Joaquín Delgado ^4,

1.	Departamento de Matemáticas y Estadística, Universidad del Norte, Barranquilla, Colombia
2.	Departamento de Matemáticas, UAM-Iztapalapa, 09340 Iztapalapa, México, D.F., Mexico
3.	Universidad Sergio Arboleda, Calle 74 no. 14-14, Bogotá, D.C.
4.	Departamento de Matemáticas, Universidad Autónoma Metropolitana – Iztapalapa, 09340 Iztapalapa, México, D. F.

Received April 2011 Revised March 2012 Published October 2012

Abstract
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In this paper we analyze the non-integrability of the Wilbeforce spring-pendulum by means of Morales-Ramis theory in where is enough to prove that the Galois group of the variational equation is not virtually abelian. We obtain these non-integrability results due to the algebrization of the variational equation falls into a Heun differential equation with four singularities and then we apply Kovacic's algorithm to determine its non-integrability.

Keywords: wilbeforce pendulum, Algebrization, hamiltonian systems, differential Galois group., Kovacic's algorithm.

Mathematics Subject Classification: Primary: 37J30; Secondary: 12H05 34M45 37J35 70G55 70H0.

Citation: Primitivo B. Acosta-Humánez, Martha Alvarez-Ramírez, David Blázquez-Sanz, Joaquín Delgado. Non-integrability criterium for normal variational equations around an integrable subsystem and an example: The Wilberforce spring-pendulum. Discrete and Continuous Dynamical Systems, 2013, 33 (3) : 965-986. doi: 10.3934/dcds.2013.33.965

References:

[1]	P. B. Acosta-Humanez, "Galoisian Approach to Supersymmetric Quantum Mechanics. The Integrability Analysis of the Schrodinger Equation by Means of Differential Galois Theory," VDM Verlag, Dr Müller, Berlin, 2010.
[2]	P. B. Acosta-Humanez, J. J. Morales-Ruiz and J. A. Weil, Galoisian approach to integrability of the schrödinger equation, Rep. Math. Phys., 67 (2011), 305-374.
[3]	R. H. Berg and T. S. Marshall, Wilberforce pendulum oscillations and normal modes, Am. J. Phys., 59 (1991), 32-38. doi: 10.1119/1.16702.
[4]	D. Blázquez-Sanz and J. J. Morales-Ruiz, Differential Galois theory of algebraic Lie-Vessiot systems, Differential algebra, complex analysis and orthogonal polynomials, Contemp. Math., Amer. Math. Soc., Providence, RI, 509 (2010), 1-58.
[5]	R. C. Churchill, J. Delgado and D. L. Rod, The spring pendulum system and the Riemann equation, New trends for Hamiltonian systems and celestial mechanics, Adv. Ser. Nonlinear Dynam., World Sci. Publ., River Edge, NJ, 8 (1996), 97-103.
[6]	J. Kovacic, An algorithm for solving second order linear homogeneous differential equations, J. Symbolic Computation, 2 (1986), 3-43. doi: 10.1016/S0747-7171(86)80010-4.
[7]	A. Maciejewski, M. Przybylska and J. A. Weil, Non-integrability of the generalized spring-pendulum problem, J. Phys. A, 37 (2004), 2579-2597.
[8]	R. Martínez and C. Simó, Non-integrability of the degenerate cases of the swinging Atwood's machine using higher order variational equations, Discrete Contin. Dyn. Syst., 29 (2011), 1-24.
[9]	J. J. Morales-Ruiz, "Differential Galois Theory and Non-integrability of Hamiltonian Systems," Progress in Mathematics 179, Birkhäuser, 1999.
[10]	J. J. Morales-Ruiz and J. P. Ramis, Galoisian obstructions to integrability of hamiltonian systems I, Methods Appl. Anal., 8 (2001), 33-95.
[11]	J. J. Morales-Ruiz and J. P. Ramis, Galoisian obstructions to integrability of hamiltonian systems II, Methods Appl. Anal., 8 (2001), 97-111.
[12]	J. J. Morales-Ruiz, C. Simó and S. Simon, Algebraic proof of the non-integrability of Hill's problem, Ergodic Theory Dynam. Systems, 25 (2005), 1237-1256.
[13]	J. J. Morales-Ruiz, J. P. Ramis and C. Simó, Integrability of hamiltonian systems and differential Galois groups of higher variational equations, Ann. Sci. École Norm. Sup. (4), 40 (2007), 845-884.
[14]	J. Muñoz, J. Rodríguez and F. J. Muriel, Weil bundles and Jet spaces, Czech. Math. J., 50 (2000), 721-748.
[15]	J. Martinet and J. P. Ramis, Théorie de Galois différentielle et resommation, Computer algebra and differential equations, Comput. Math. Appl., Academic Press, London, (1990), 117-224.

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References:

[1]	P. B. Acosta-Humanez, "Galoisian Approach to Supersymmetric Quantum Mechanics. The Integrability Analysis of the Schrodinger Equation by Means of Differential Galois Theory," VDM Verlag, Dr Müller, Berlin, 2010.
[2]	P. B. Acosta-Humanez, J. J. Morales-Ruiz and J. A. Weil, Galoisian approach to integrability of the schrödinger equation, Rep. Math. Phys., 67 (2011), 305-374.
[3]	R. H. Berg and T. S. Marshall, Wilberforce pendulum oscillations and normal modes, Am. J. Phys., 59 (1991), 32-38. doi: 10.1119/1.16702.
[4]	D. Blázquez-Sanz and J. J. Morales-Ruiz, Differential Galois theory of algebraic Lie-Vessiot systems, Differential algebra, complex analysis and orthogonal polynomials, Contemp. Math., Amer. Math. Soc., Providence, RI, 509 (2010), 1-58.
[5]	R. C. Churchill, J. Delgado and D. L. Rod, The spring pendulum system and the Riemann equation, New trends for Hamiltonian systems and celestial mechanics, Adv. Ser. Nonlinear Dynam., World Sci. Publ., River Edge, NJ, 8 (1996), 97-103.
[6]	J. Kovacic, An algorithm for solving second order linear homogeneous differential equations, J. Symbolic Computation, 2 (1986), 3-43. doi: 10.1016/S0747-7171(86)80010-4.
[7]	A. Maciejewski, M. Przybylska and J. A. Weil, Non-integrability of the generalized spring-pendulum problem, J. Phys. A, 37 (2004), 2579-2597.
[8]	R. Martínez and C. Simó, Non-integrability of the degenerate cases of the swinging Atwood's machine using higher order variational equations, Discrete Contin. Dyn. Syst., 29 (2011), 1-24.
[9]	J. J. Morales-Ruiz, "Differential Galois Theory and Non-integrability of Hamiltonian Systems," Progress in Mathematics 179, Birkhäuser, 1999.
[10]	J. J. Morales-Ruiz and J. P. Ramis, Galoisian obstructions to integrability of hamiltonian systems I, Methods Appl. Anal., 8 (2001), 33-95.
[11]	J. J. Morales-Ruiz and J. P. Ramis, Galoisian obstructions to integrability of hamiltonian systems II, Methods Appl. Anal., 8 (2001), 97-111.
[12]	J. J. Morales-Ruiz, C. Simó and S. Simon, Algebraic proof of the non-integrability of Hill's problem, Ergodic Theory Dynam. Systems, 25 (2005), 1237-1256.
[13]	J. J. Morales-Ruiz, J. P. Ramis and C. Simó, Integrability of hamiltonian systems and differential Galois groups of higher variational equations, Ann. Sci. École Norm. Sup. (4), 40 (2007), 845-884.
[14]	J. Muñoz, J. Rodríguez and F. J. Muriel, Weil bundles and Jet spaces, Czech. Math. J., 50 (2000), 721-748.
[15]	J. Martinet and J. P. Ramis, Théorie de Galois différentielle et resommation, Computer algebra and differential equations, Comput. Math. Appl., Academic Press, London, (1990), 117-224.

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