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Journal of Mathematical Physics / Volume 52 / Issue 3 / ARTICLES / Classical Mechanics and Classical Fields

J. Math. Phys. 52, 032901 (2011); http://dx.doi.org/10.1063/1.3559145 (9 pages)

Periodic orbits and nonintegrability of generalized classical Yang–Mills Hamiltonian systems

Lidia Jiménez–Lara1 and Jaume Llibre2

1Departamento de Física, Universidad Autónoma Metropolitana–Iztapalapa, P.O. Box 55–534, México, D.F., 09340 México
2Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain

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(Received 22 August 2010; accepted 2 February 2011; published online 2 March 2011)

The averaging theory of first order is applied to study a generalized Yang–Mills system with two parameters. Two main results are proved. First, we provide sufficient conditions on the two parameters of the generalized system to guarantee the existence of continuous families of isolated periodic orbits parameterized by the energy, and these families are given up to first order in a small parameter. Second, we prove that for the nonintegrable classical Yang–Mills Hamiltonian systems, in the sense of Liouville–Arnold, which have the isolated periodic orbits found with averaging theory, cannot exist in any second first integral of class C1. This is important because most of the results about integrability deals with analytic or meromorphic integrals of motion.

© 2011 American Institute of Physics

Article Outline

  1. INTRODUCTION
  2. THE AVERAGING THEORY
  3. PROOF OF THEOREM 1
  4. PERIODIC ORBITS AND THE LIOUVILLE–ARNOLD INTEGRABILITY
  5. CONCLUSIONS

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KEYWORDS and PACS

Keywords

differential equations, harmonic oscillators, Yang-Mills theory

PACS

  • 03.65.Ge

    Solutions of wave equations: bound states

  • 11.15.-q

    Gauge field theories

  • 02.30.Jr

    Partial differential equations

ARTICLE DATA

Digital Object Identifier
http://dx.doi.org/10.1063/1.3559145

PUBLICATION DATA

ISSN

0022-2488 (print)  
1089-7658 (online)

Publisher

$abstract.society.value is a Member of CrossRef American Institute of Physics

For access to fully linked references, you need to log in.
    Bountis, T., Segur, H., and Vivaldi, F., “Integrable Hamiltonian systems and the Painlevé propety,” Phys. Rev. A, 25, 1257 (1982).

    Dorizzi, B., Grammaticos, B., and Ramani, A., “A new class of integrable systems,” J. Math. Phys. 24, 2282 (1983)JMAPAQ000024000009002282000001.

    Friedberg, R., Lee, T., and Sirlin, A., “Class of scalar-field soliton solutions in three space dimensions,” Phys. Rev. D 13, 2739 (1976).

    Grammaticos, B., Dorizzi, B., and Ramani, A., “Integrability of Hamiltonians with third- and fourth–degree polynomial potentials,” J. Math. Phys. 24, 2289 (1983)JMAPAQ000024000009002289000001.

    Ramani, A., Dorizzi, B., and Grammaticos, B., “Painlevé conjecture revisited,” Phys. Rev. Lett. 49, 1539 (1982).


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