American Institute of Mathematical Sciences

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April  2013, 33(4): 1645-1655. doi: 10.3934/dcds.2013.33.1645

Non-integrability of generalized Yang-Mills Hamiltonian system

Shaoyun Shi 1, and  Wenlei Li 2,

1. 

College of Mathematics, & Key Laboratory of Symbolic Computation and Knowledge Engineering of Ministry of Education, Jilin University, Changchun 130012, China
2. 

College of Mathematics, Jilin University, Changchun 130012, China

Received  November 2011 Revised  May 2012 Published  October 2012

We show that the generalized Yang-Mills system with Hamiltonian $H=\frac12(y_1^2+y_2^2)+\frac12(ax_1^2+bx_2^2)+\frac14cx_1^4+\frac14dx_2^4+\frac12ex_1^2x_2^2$ is meromorphically integrable in Liouvillian sense(i.e., the existence of an additional meromorphic first integral) if and only if (A) $e=0$, or (B) $c=d=e$, or (C) $a=b, e=3c=3d$, or (D) $b=4a, e=3c, d=8c$, or (E) $b=4a, e=6c, d=16c$, or (F) $b=4a, e=3d, c=8d$, or (G) $b=4a, e=6d, c=16d$. Therefore, we get a complete classification of the Yang-Mills Hamiltonian system in sense of integrability and non-integrability.
Keywords: Non-integrability, Morales-Ramis theory, higher order variational equations., Lamé equation, Yang-Mills system.
Mathematics Subject Classification: Primary: 70H06, 34M15; Secondary: 34M0.
Citation: Shaoyun Shi, Wenlei Li. Non-integrability of generalized Yang-Mills Hamiltonian system. Discrete & Continuous Dynamical Systems, 2013, 33 (4) : 1645-1655. doi: 10.3934/dcds.2013.33.1645
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show all references

References:
[1]

Nonlinear Studies The International Journal, 16 (2009), 299-314.  Google Scholar

[2]

Fields Inst. Commun., 7 (1996), 5-56.  Google Scholar

[3]

J. Differential Equations, 41 (1981), 44-58.  Google Scholar

[4]

Proc. R. Soc. Lond. A, 434 (1991), 263-278.  Google Scholar

[5]

Brazilian Journal of Physics, 37 (2007), 398-405. doi: 10.1007/s10765-007-0152-8.  Google Scholar

[6]

Phys. Rev. A., 25 (1982), 1257-1264.  Google Scholar

[7]

Ergod. Th & Dynam. Sys. (1), 5 (1995), 15-48.  Google Scholar

[8]

Brazilian Journal of Physics, 35 (2005). Google Scholar

[9]

Amer. J. Math., 112 (1990), 749-786. doi: 10.2307/2374806.  Google Scholar

[10]

Celest. Mech. Dynam. Astr., 62 (1995), 191-192. doi: 10.1007/BF00692087.  Google Scholar

[11]

Phys. Rev. D., 13 (1976), 2739-2761.  Google Scholar

[12]

Gauthier-Villars, Paris, (1888). Google Scholar

[13]

Phys. Rep., 147 (1987), 87-154. doi: 10.1016/0370-1573(87)90089-5.  Google Scholar

[14]

Celest. Mech. Dynam. Astr., 58 (1994), 387-391.  Google Scholar

[15]

Celest. Mech. Dynam. Astr., 109 (2010), 1-12.  Google Scholar

[16]

J. Phys. A., 41 (2008), 26 pp. 465101.  Google Scholar

[17]

J. Phys. A., 37 (2004), 2579-2597.  Google Scholar

[18]

J. Math. Phys., 46 (2005), 062901.  Google Scholar

[19]

Soviet Phys. JETP., 38 (1974), 248-253. Google Scholar

[20]

Ph.D. Thesis, University of Barcelona, 1989. Google Scholar

[21]

J. Differential Equations, 107 (1994), 140-162.  Google Scholar

[22]

Birkhäuser Verlag, Basel, 1999.  Google Scholar

[23]

J. Differential Equations, 129 (1996), 111-135.  Google Scholar

[24]

Ergod. Th & Dynam. Sys., 25 (2005), 1237-1256.  Google Scholar

[25]

Annales Scientifiques de l'école Normale Supéieure, 40 (2007), 845-884.  Google Scholar

[26]

Discrete Contin. Dyn. Syst., 24 (2009), 1225-1273.  Google Scholar

[27]

Oxford Univ. Press, London, 1936. Google Scholar

[28]

Phys. Rev. D., 11 (1975), 2950-2966. Google Scholar

[29]

volume 328 of Grundlehren der mathematischen Wissenshaften. Springer. Heidelberg, 2003.  Google Scholar

[30]

Compositio Math., 29 (1994), 157-203. doi: 10.1016/0165-0270(94)90123-6.  Google Scholar

[31]

Cambrige Univ. Press, Cambrige, 1969. Google Scholar

[32]

Zh. Eksp. Teor. Fiz. Lett., 30 (1979), 39-44. Google Scholar

[33]

Funct. Anal. Appl., 16 (1983), 181-189; 6-17. Google Scholar

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