# American Institute of Mathematical Sciences

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

## Non-integrability of generalized Yang-Mills Hamiltonian system

 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.
Citation: Shaoyun Shi, Wenlei Li. Non-integrability of generalized Yang-Mills Hamiltonian system. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1645-1655. doi: 10.3934/dcds.2013.33.1645
##### References:
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Vivaldi, Integrable Hamiltonian systems and the Painleve property,, Phys. Rev. A., 25 (1982), 1257. Google Scholar [7] R. C. Churchill, D. L. Rod and M. F. Singer, Group-theoretic obstructions to integrability,, Ergod. Th & Dynam. Sys. (1), 5 (1995), 15. Google Scholar [8] L. A. A. Cohelo, J. E. F. Skea and T. J. Stuchi, On the non-integrability of a class of Hamiltonian cosmological models,, Brazilian Journal of Physics, 35 (2005). Google Scholar [9] B. Dwork, Differential operators with nilponent $p$-curvature,, Amer. J. Math., 112 (1990), 749. doi: 10.2307/2374806. Google Scholar [10] A. Elipe, J. Hietarinta and S. Tompaidis, Comment on paper by S. Kasperczuk, Celest. Mech 58:387-391(1994),, Celest. Mech. Dynam. Astr., 62 (1995), 191. doi: 10.1007/BF00692087. Google Scholar [11] R. Fridberg, T. D. Lee and R. Padjen, Class of scalar-field solutions in three space dimensions,, Phys. Rev. D., 13 (1976), 2739. Google Scholar [12] G. H. Halphen, Traité des fonctions elliptiques VOl. I, II,, Gauthier-Villars, (1888). Google Scholar [13] J. Hietarinta, Direct methods for the search of the second invariant,, Phys. Rep., 147 (1987), 87. doi: 10.1016/0370-1573(87)90089-5. Google Scholar [14] S. Kasperczuk, Integrability of the Yang-Mills Hamiltonian system,, Celest. Mech. Dynam. Astr., 58 (1994), 387. Google Scholar [15] W. L. Li and S. Y. Shi, Non-integrability of Hénon-Heiles System,, Celest. Mech. Dynam. Astr., 109 (2010), 1. Google Scholar [16] A. J. Maciejewski, M. Przybylska, T. Stachowiak and M. Szydlowski, Global integrability of cosmological scalar fields,, J. Phys. A., 41 (2008). Google Scholar [17] A. J. Maciejewski, M. Przybylska and J. A. Weil, Non-integrability of the generalized spring-pendulum problem,, J. Phys. A., 37 (2004), 2579. Google Scholar [18] A. J. Maciejewski and M. Przybylska, Darboux points and integrability of Hamiltonian systems with homogeneous polynomial potential,, J. Math. Phys., 46 (2005). Google Scholar [19] S. V. Manakov, On the theory of two-dimensional stationary self-focusing of electromagnetic waves,, Soviet Phys. JETP., 38 (1974), 248. Google Scholar [20] J. J. Morales-Ruiz, "Técnicas Algebraicas Para el Estudio de la Integrabilidad de Sistemas Hamiltonianos,", Ph.D. Thesis, (1989). Google Scholar [21] J. J. Morales-Ruiz and C. Simó, Picard-Vessiot theory and Ziglin's theory,, J. Differential Equations, 107 (1994), 140. Google Scholar [22] J. J. Morales-Ruiz, "Differential Galois Theory and Non-Integrability of Hamiltonian Systems,", Birkhäuser Verlag, (1999). Google Scholar [23] J. J. Morales-Ruiz and C. Simó, Non-integrability criteria for Hamiltonians in the case of Lamé normal variational equations,, J. Differential Equations, 129 (1996), 111. Google Scholar [24] J. J. Morales-Ruiz, C. Simó and S. Simon, Algebraic proof of the non-integrability of Hill's problem,, Ergod. Th & Dynam. Sys., 25 (2005), 1237. Google Scholar [25] J. J. Morales-Ruiz, J. P. Ramis and C. Simó, Integrability of Hamiltonian systems and differential Galois groups of higher variational equations,, Annales Scientifiques de l'école Normale Supéieure, 40 (2007), 845. Google Scholar [26] J. J. Morales-Ruiz and S. Simon, On the meromorphic non-integrability of some $N$-body problems,, Discrete Contin. Dyn. Syst., 24 (2009), 1225. Google Scholar [27] E. G. C. Poole, "Introduction to the Theory of Linear Differential Equations,", Oxford Univ. Press, (1936). Google Scholar [28] R. Rajaraman and E. J. Weinberg, Internal symmetry and the semi-classical method in quantum field theory,, Phys. Rev. D., 11 (1975), 2950. Google Scholar [29] Van der Put M and M. F. Singer, "Galois Theory of Linear Differential Equations,", volume 328 of Grundlehren der mathematischen Wissenshaften. Springer. Heidelberg, (2003). Google Scholar [30] P. Vanhaecke, A special case of the Garnier system, (1,4)-polarised Abelian surfaces and their moduli,, Compositio Math., 29 (1994), 157. doi: 10.1016/0165-0270(94)90123-6. Google Scholar [31] E. T. Whittaker and E. T. Watson, "A Course of Modern Analysis,", Cambrige Univ. Press, (1969). Google Scholar [32] V. E. Zakharv, M. F. Ivanov and L. I. Shoor, On anomalously slow stochastization in certain two-dimensional models of field theory,, Zh. Eksp. Teor. Fiz. Lett., 30 (1979), 39. Google Scholar [33] S. L. Ziglin, Branching of solutions and non-existence of first integrals in Hamiltonian mechanics I, II,, Funct. Anal. Appl., 16 (1983), 181. Google Scholar

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##### References:
 [1] P. B. Acosta-Humanez, D. Blazquez-Sanz and C. V. Contreras, On Hamiltonian potentials with quartic polynomial normal variational equations,, Nonlinear Studies The International Journal, 16 (2009), 299. Google Scholar [2] A. Baider, R. C. Churchill, D. L. Rod and M. F. Singer, On the infinitesimal geometry of integrable systems,, Fields Inst. Commun., 7 (1996), 5. Google Scholar [3] F. Baldassarri, On Algebraic solution of Lamé's differential equation,, J. Differential Equations, 41 (1981), 44. Google Scholar [4] G. Baumann, W. G. Glöckle and T. F. Nonnenmacher, Sigular point analysis and integrals of motion for coupled nonlinear Schrödinger equations,, Proc. R. Soc. Lond. A, 434 (1991), 263. Google Scholar [5] D. Boucher and J. A. Weil, About nonintegrability in the Friedmann-Robertson-Walker cosmological model,, Brazilian Journal of Physics, 37 (2007), 398. doi: 10.1007/s10765-007-0152-8. Google Scholar [6] T. Bountis, H. Segur and F. Vivaldi, Integrable Hamiltonian systems and the Painleve property,, Phys. Rev. A., 25 (1982), 1257. Google Scholar [7] R. C. Churchill, D. L. Rod and M. F. Singer, Group-theoretic obstructions to integrability,, Ergod. Th & Dynam. Sys. (1), 5 (1995), 15. Google Scholar [8] L. A. A. Cohelo, J. E. F. Skea and T. J. Stuchi, On the non-integrability of a class of Hamiltonian cosmological models,, Brazilian Journal of Physics, 35 (2005). Google Scholar [9] B. Dwork, Differential operators with nilponent $p$-curvature,, Amer. J. Math., 112 (1990), 749. doi: 10.2307/2374806. Google Scholar [10] A. Elipe, J. Hietarinta and S. Tompaidis, Comment on paper by S. Kasperczuk, Celest. Mech 58:387-391(1994),, Celest. Mech. Dynam. Astr., 62 (1995), 191. doi: 10.1007/BF00692087. Google Scholar [11] R. Fridberg, T. D. Lee and R. Padjen, Class of scalar-field solutions in three space dimensions,, Phys. Rev. D., 13 (1976), 2739. Google Scholar [12] G. H. Halphen, Traité des fonctions elliptiques VOl. I, II,, Gauthier-Villars, (1888). Google Scholar [13] J. Hietarinta, Direct methods for the search of the second invariant,, Phys. Rep., 147 (1987), 87. doi: 10.1016/0370-1573(87)90089-5. Google Scholar [14] S. Kasperczuk, Integrability of the Yang-Mills Hamiltonian system,, Celest. Mech. Dynam. Astr., 58 (1994), 387. Google Scholar [15] W. L. Li and S. Y. Shi, Non-integrability of Hénon-Heiles System,, Celest. Mech. Dynam. Astr., 109 (2010), 1. Google Scholar [16] A. J. Maciejewski, M. Przybylska, T. Stachowiak and M. Szydlowski, Global integrability of cosmological scalar fields,, J. Phys. A., 41 (2008). Google Scholar [17] A. J. Maciejewski, M. Przybylska and J. A. Weil, Non-integrability of the generalized spring-pendulum problem,, J. Phys. A., 37 (2004), 2579. Google Scholar [18] A. J. Maciejewski and M. Przybylska, Darboux points and integrability of Hamiltonian systems with homogeneous polynomial potential,, J. Math. Phys., 46 (2005). Google Scholar [19] S. V. Manakov, On the theory of two-dimensional stationary self-focusing of electromagnetic waves,, Soviet Phys. JETP., 38 (1974), 248. Google Scholar [20] J. J. Morales-Ruiz, "Técnicas Algebraicas Para el Estudio de la Integrabilidad de Sistemas Hamiltonianos,", Ph.D. Thesis, (1989). Google Scholar [21] J. J. Morales-Ruiz and C. Simó, Picard-Vessiot theory and Ziglin's theory,, J. Differential Equations, 107 (1994), 140. Google Scholar [22] J. J. Morales-Ruiz, "Differential Galois Theory and Non-Integrability of Hamiltonian Systems,", Birkhäuser Verlag, (1999). Google Scholar [23] J. J. Morales-Ruiz and C. Simó, Non-integrability criteria for Hamiltonians in the case of Lamé normal variational equations,, J. Differential Equations, 129 (1996), 111. Google Scholar [24] J. J. Morales-Ruiz, C. Simó and S. Simon, Algebraic proof of the non-integrability of Hill's problem,, Ergod. Th & Dynam. Sys., 25 (2005), 1237. Google Scholar [25] J. J. Morales-Ruiz, J. P. Ramis and C. Simó, Integrability of Hamiltonian systems and differential Galois groups of higher variational equations,, Annales Scientifiques de l'école Normale Supéieure, 40 (2007), 845. Google Scholar [26] J. J. Morales-Ruiz and S. Simon, On the meromorphic non-integrability of some $N$-body problems,, Discrete Contin. Dyn. Syst., 24 (2009), 1225. Google Scholar [27] E. G. C. Poole, "Introduction to the Theory of Linear Differential Equations,", Oxford Univ. Press, (1936). Google Scholar [28] R. Rajaraman and E. J. Weinberg, Internal symmetry and the semi-classical method in quantum field theory,, Phys. Rev. D., 11 (1975), 2950. Google Scholar [29] Van der Put M and M. F. Singer, "Galois Theory of Linear Differential Equations,", volume 328 of Grundlehren der mathematischen Wissenshaften. Springer. Heidelberg, (2003). Google Scholar [30] P. Vanhaecke, A special case of the Garnier system, (1,4)-polarised Abelian surfaces and their moduli,, Compositio Math., 29 (1994), 157. doi: 10.1016/0165-0270(94)90123-6. Google Scholar [31] E. T. Whittaker and E. T. Watson, "A Course of Modern Analysis,", Cambrige Univ. Press, (1969). Google Scholar [32] V. E. Zakharv, M. F. Ivanov and L. I. Shoor, On anomalously slow stochastization in certain two-dimensional models of field theory,, Zh. Eksp. Teor. Fiz. Lett., 30 (1979), 39. Google Scholar [33] S. L. Ziglin, Branching of solutions and non-existence of first integrals in Hamiltonian mechanics I, II,, Funct. Anal. Appl., 16 (1983), 181. Google Scholar
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