American Institute of Mathematical Sciences

Advanced
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

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 & 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]

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
[2]

Simon Hochgerner. Symmetry actuated closed-loop Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 641-669. doi: 10.3934/jgm.2020030
[3]

Hua Shi, Xiang Zhang, Yuyan Zhang. Complex planar Hamiltonian systems: Linearization and dynamics. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020406
[4]

Manuel de León, Víctor M. Jiménez, Manuel Lainz. Contact Hamiltonian and Lagrangian systems with nonholonomic constraints. Journal of Geometric Mechanics, 2020  doi: 10.3934/jgm.2021001
[5]

João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138
[6]

Adrian Viorel, Cristian D. Alecsa, Titus O. Pinţa. Asymptotic analysis of a structure-preserving integrator for damped Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020407
[7]

Guo Zhou, Yongquan Zhou, Ruxin Zhao. Hybrid social spider optimization algorithm with differential mutation operator for the job-shop scheduling problem. Journal of Industrial & Management Optimization, 2021, 17 (2) : 533-548. doi: 10.3934/jimo.2019122
[8]

Huanhuan Tian, Maoan Han. Limit cycle bifurcations of piecewise smooth near-Hamiltonian systems with a switching curve. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020368
[9]

Ying Lv, Yan-Fang Xue, Chun-Lei Tang. Ground state homoclinic orbits for a class of asymptotically periodic second-order Hamiltonian systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1627-1652. doi: 10.3934/dcdsb.2020176
[10]

Lan Luo, Zhe Zhang, Yong Yin. Simulated annealing and genetic algorithm based method for a bi-level seru loading problem with worker assignment in seru production systems. Journal of Industrial & Management Optimization, 2021, 17 (2) : 779-803. doi: 10.3934/jimo.2019134
[11]

Qingfeng Zhu, Yufeng Shi. Nonzero-sum differential game of backward doubly stochastic systems with delay and applications. Mathematical Control & Related Fields, 2021, 11 (1) : 73-94. doi: 10.3934/mcrf.2020028
[12]

Xianwei Chen, Xiangling Fu, Zhujun Jing. Chaos control in a special pendulum system for ultra-subharmonic resonance. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 847-860. doi: 10.3934/dcdsb.2020144
[13]

Laurent Di Menza, Virginie Joanne-Fabre. An age group model for the study of a population of trees. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020464
[14]

Qiao Liu. Local rigidity of certain solvable group actions on tori. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 553-567. doi: 10.3934/dcds.2020269
[15]

Kien Trung Nguyen, Vo Nguyen Minh Hieu, Van Huy Pham. Inverse group 1-median problem on trees. Journal of Industrial & Management Optimization, 2021, 17 (1) : 221-232. doi: 10.3934/jimo.2019108
[16]

Meihua Dong, Keonhee Lee, Carlos Morales. Gromov-Hausdorff stability for group actions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1347-1357. doi: 10.3934/dcds.2020320
[17]

Wolfgang Riedl, Robert Baier, Matthias Gerdts. Optimization-based subdivision algorithm for reachable sets. Journal of Computational Dynamics, 2021, 8 (1) : 99-130. doi: 10.3934/jcd.2021005
[18]

Hong Fu, Mingwu Liu, Bo Chen. Supplier's investment in manufacturer's quality improvement with equity holding. Journal of Industrial & Management Optimization, 2021, 17 (2) : 649-668. doi: 10.3934/jimo.2019127
[19]

Skyler Simmons. Stability of broucke's isosceles orbit. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021015
[20]

François Ledrappier. Three problems solved by Sébastien Gouëzel. Journal of Modern Dynamics, 2020, 16: 373-387. doi: 10.3934/jmd.2020015

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (96)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences