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

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

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.
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 & Continuous Dynamical Systems - A, 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, (2010).   Google Scholar

[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.   Google Scholar

[3]

R. H. Berg and T. S. Marshall, Wilberforce pendulum oscillations and normal modes,, Am. J. Phys., 59 (1991), 32.  doi: 10.1119/1.16702.  Google Scholar

[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., 509 (2010), 1.   Google Scholar

[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, 8 (1996), 97.   Google Scholar

[6]

J. Kovacic, An algorithm for solving second order linear homogeneous differential equations,, J. Symbolic Computation, 2 (1986), 3.  doi: 10.1016/S0747-7171(86)80010-4.  Google Scholar

[7]

A. Maciejewski, M. Przybylska and J. A. Weil, Non-integrability of the generalized spring-pendulum problem,, J. Phys. A, 37 (2004), 2579.   Google Scholar

[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.   Google Scholar

[9]

J. J. Morales-Ruiz, "Differential Galois Theory and Non-integrability of Hamiltonian Systems,", Progress in Mathematics 179, (1999).   Google Scholar

[10]

J. J. Morales-Ruiz and J. P. Ramis, Galoisian obstructions to integrability of hamiltonian systems I,, Methods Appl. Anal., 8 (2001), 33.   Google Scholar

[11]

J. J. Morales-Ruiz and J. P. Ramis, Galoisian obstructions to integrability of hamiltonian systems II,, Methods Appl. Anal., 8 (2001), 97.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[14]

J. Muñoz, J. Rodríguez and F. J. Muriel, Weil bundles and Jet spaces,, Czech. Math. J., 50 (2000), 721.   Google Scholar

[15]

J. Martinet and J. P. Ramis, Théorie de Galois différentielle et resommation,, Computer algebra and differential equations, (1990), 117.   Google Scholar

show all references

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, (2010).   Google Scholar

[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.   Google Scholar

[3]

R. H. Berg and T. S. Marshall, Wilberforce pendulum oscillations and normal modes,, Am. J. Phys., 59 (1991), 32.  doi: 10.1119/1.16702.  Google Scholar

[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., 509 (2010), 1.   Google Scholar

[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, 8 (1996), 97.   Google Scholar

[6]

J. Kovacic, An algorithm for solving second order linear homogeneous differential equations,, J. Symbolic Computation, 2 (1986), 3.  doi: 10.1016/S0747-7171(86)80010-4.  Google Scholar

[7]

A. Maciejewski, M. Przybylska and J. A. Weil, Non-integrability of the generalized spring-pendulum problem,, J. Phys. A, 37 (2004), 2579.   Google Scholar

[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.   Google Scholar

[9]

J. J. Morales-Ruiz, "Differential Galois Theory and Non-integrability of Hamiltonian Systems,", Progress in Mathematics 179, (1999).   Google Scholar

[10]

J. J. Morales-Ruiz and J. P. Ramis, Galoisian obstructions to integrability of hamiltonian systems I,, Methods Appl. Anal., 8 (2001), 33.   Google Scholar

[11]

J. J. Morales-Ruiz and J. P. Ramis, Galoisian obstructions to integrability of hamiltonian systems II,, Methods Appl. Anal., 8 (2001), 97.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[14]

J. Muñoz, J. Rodríguez and F. J. Muriel, Weil bundles and Jet spaces,, Czech. Math. J., 50 (2000), 721.   Google Scholar

[15]

J. Martinet and J. P. Ramis, Théorie de Galois différentielle et resommation,, Computer algebra and differential equations, (1990), 117.   Google Scholar

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