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, 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. 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), 305-374.  Google Scholar

[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.  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., Amer. Math. Soc., Providence, RI, 509 (2010), 1-58.  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, Adv. Ser. Nonlinear Dynam., World Sci. Publ., River Edge, NJ, 8 (1996), 97-103.  Google Scholar

[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.  Google Scholar

[7]

A. Maciejewski, M. Przybylska and J. A. Weil, Non-integrability of the generalized spring-pendulum problem, J. Phys. A, 37 (2004), 2579-2597.  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), 1-24.  Google Scholar

[9]

J. J. Morales-Ruiz, "Differential Galois Theory and Non-integrability of Hamiltonian Systems," Progress in Mathematics 179, Birkhäuser, 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), 33-95.  Google Scholar

[11]

J. J. Morales-Ruiz and J. P. Ramis, Galoisian obstructions to integrability of hamiltonian systems II, Methods Appl. Anal., 8 (2001), 97-111. 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), 1237-1256.  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), 845-884.  Google Scholar

[14]

J. Muñoz, J. Rodríguez and F. J. Muriel, Weil bundles and Jet spaces, Czech. Math. J., 50 (2000), 721-748.  Google Scholar

[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.  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, Dr Müller, Berlin, 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), 305-374.  Google Scholar

[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.  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., Amer. Math. Soc., Providence, RI, 509 (2010), 1-58.  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, Adv. Ser. Nonlinear Dynam., World Sci. Publ., River Edge, NJ, 8 (1996), 97-103.  Google Scholar

[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.  Google Scholar

[7]

A. Maciejewski, M. Przybylska and J. A. Weil, Non-integrability of the generalized spring-pendulum problem, J. Phys. A, 37 (2004), 2579-2597.  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), 1-24.  Google Scholar

[9]

J. J. Morales-Ruiz, "Differential Galois Theory and Non-integrability of Hamiltonian Systems," Progress in Mathematics 179, Birkhäuser, 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), 33-95.  Google Scholar

[11]

J. J. Morales-Ruiz and J. P. Ramis, Galoisian obstructions to integrability of hamiltonian systems II, Methods Appl. Anal., 8 (2001), 97-111. 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), 1237-1256.  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), 845-884.  Google Scholar

[14]

J. Muñoz, J. Rodríguez and F. J. Muriel, Weil bundles and Jet spaces, Czech. Math. J., 50 (2000), 721-748.  Google Scholar

[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.  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, 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, 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]

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

[5]

Sonja Hohloch, Silvia Sabatini, Daniele Sepe. From compact semi-toric systems to Hamiltonian $S^1$-spaces. Discrete & Continuous Dynamical Systems, 2015, 35 (1) : 247-281. doi: 10.3934/dcds.2015.35.247

[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, 2013, 33 (3) : 1201-1214. doi: 10.3934/dcds.2013.33.1201

[8]

Antonio Giorgilli. Unstable equilibria of Hamiltonian systems. Discrete & Continuous Dynamical Systems, 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, 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, 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, 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]

Jean-François Biasse, Muhammed Rashad Erukulangara. A proof of the conjectured run time of the Hafner-McCurley class group algorithm. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021055

[16]

Sergey Rashkovskiy. Hamilton-Jacobi theory for Hamiltonian and non-Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 563-583. doi: 10.3934/jgm.2020024

[17]

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

[18]

Alexander Moreto. Complex group algebras of finite groups: Brauer's Problem 1. Electronic Research Announcements, 2005, 11: 34-39.

[19]

Ammari Zied, Liard Quentin. On uniqueness of measure-valued solutions to Liouville's equation of Hamiltonian PDEs. Discrete & Continuous Dynamical Systems, 2018, 38 (2) : 723-748. doi: 10.3934/dcds.2018032

[20]

K. Tintarev. Critical values and minimal periods for autonomous Hamiltonian systems. Discrete & Continuous Dynamical Systems, 1995, 1 (3) : 389-400. doi: 10.3934/dcds.1995.1.389

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (121)
  • HTML views (0)
  • Cited by (3)

[Back to Top]