# American Institute of Mathematical Sciences

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
 [1] David Blázquez-Sanz, Juan J. Morales-Ruiz. Lie's reduction method and differential Galois theory in the complex analytic context. Discrete & Continuous Dynamical Systems - A, 2012, 32 (2) : 353-379. doi: 10.3934/dcds.2012.32.353 [2] Alicia Cordero, José Martínez Alfaro, Pura Vindel. Bott integrable Hamiltonian systems on $S^{2}\times S^{1}$. Discrete & Continuous Dynamical Systems - A, 2008, 22 (3) : 587-604. doi: 10.3934/dcds.2008.22.587 [3] Jan J. Sławianowski, Vasyl Kovalchuk, Agnieszka Martens, Barbara Gołubowska, Ewa E. Rożko. Essential nonlinearity implied by symmetry group. Problems of affine invariance in mechanics and physics. Discrete & Continuous Dynamical Systems - B, 2012, 17 (2) : 699-733. doi: 10.3934/dcdsb.2012.17.699 [4] Sonja Hohloch, Silvia Sabatini, Daniele Sepe. From compact semi-toric systems to Hamiltonian $S^1$-spaces. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 247-281. doi: 10.3934/dcds.2015.35.247 [5] Roman Srzednicki. On periodic solutions in the Whitney's inverted pendulum problem. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2127-2141. doi: 10.3934/dcdss.2019137 [6] Tomasz Kapela, Piotr Zgliczyński. A Lohner-type algorithm for control systems and ordinary differential inclusions. Discrete & Continuous Dynamical Systems - B, 2009, 11 (2) : 365-385. doi: 10.3934/dcdsb.2009.11.365 [7] Kenneth R. Meyer, Jesús F. Palacián, Patricia Yanguas. Normally stable hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (3) : 1201-1214. doi: 10.3934/dcds.2013.33.1201 [8] Antonio Giorgilli. Unstable equilibria of Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2001, 7 (4) : 855-871. doi: 10.3934/dcds.2001.7.855 [9] Dongfeng Zhang, Junxiang Xu. On elliptic lower dimensional tori for Gevrey-smooth Hamiltonian systems under Rüssmann's non-degeneracy condition. Discrete & Continuous Dynamical Systems - A, 2006, 16 (3) : 635-655. doi: 10.3934/dcds.2006.16.635 [10] John Fogarty. On Noether's bound for polynomial invariants of a finite group. Electronic Research Announcements, 2001, 7: 5-7. [11] Xiaojun Huang, Jinsong Liu, Changrong Zhu. The Katok's entropy formula for amenable group actions. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4467-4482. doi: 10.3934/dcds.2018195 [12] Martin Pinsonnault. Maximal compact tori in the Hamiltonian group of 4-dimensional symplectic manifolds. Journal of Modern Dynamics, 2008, 2 (3) : 431-455. doi: 10.3934/jmd.2008.2.431 [13] Nobuyuki Kato, Norio Kikuchi. Campanato-type boundary estimates for Rothe's scheme to parabolic partial differential systems with constant coefficients. Discrete & Continuous Dynamical Systems - A, 2007, 19 (4) : 737-760. doi: 10.3934/dcds.2007.19.737 [14] Edward Hooton, Pavel Kravetc, Dmitrii Rachinskii, Qingwen Hu. Selective Pyragas control of Hamiltonian systems. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2019-2034. doi: 10.3934/dcdss.2019130 [15] Crnković Dean, Vedrana Mikulić Crnković, Bernardo G. Rodrigues. On self-orthogonal designs and codes related to Held's simple group. Advances in Mathematics of Communications, 2018, 12 (3) : 607-628. doi: 10.3934/amc.2018036 [16] Alexander Moreto. Complex group algebras of finite groups: Brauer's Problem 1. Electronic Research Announcements, 2005, 11: 34-39. [17] Ammari Zied, Liard Quentin. On uniqueness of measure-valued solutions to Liouville's equation of Hamiltonian PDEs. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 723-748. doi: 10.3934/dcds.2018032 [18] K. Tintarev. Critical values and minimal periods for autonomous Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 1995, 1 (3) : 389-400. doi: 10.3934/dcds.1995.1.389 [19] Rumei Zhang, Jin Chen, Fukun Zhao. Multiple solutions for superlinear elliptic systems of Hamiltonian type. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1249-1262. doi: 10.3934/dcds.2011.30.1249 [20] Tianqing An, Zhi-Qiang Wang. Periodic solutions of Hamiltonian systems with anisotropic growth. Communications on Pure & Applied Analysis, 2010, 9 (4) : 1069-1082. doi: 10.3934/cpaa.2010.9.1069

2018 Impact Factor: 1.143