Non-integrability of generalized Yang-Mills Hamiltonian system

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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

Abstract
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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 and Continuous Dynamical Systems, 2013, 33 (4) : 1645-1655. doi: 10.3934/dcds.2013.33.1645

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), 299-314.
[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), 5-56.
[3]	F. Baldassarri, On Algebraic solution of Lamé's differential equation, J. Differential Equations, 41 (1981), 44-58.
[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), 263-278.
[5]	D. Boucher and J. A. Weil, About nonintegrability in the Friedmann-Robertson-Walker cosmological model, Brazilian Journal of Physics, 37 (2007), 398-405. doi: 10.1007/s10765-007-0152-8.
[6]	T. Bountis, H. Segur and F. Vivaldi, Integrable Hamiltonian systems and the Painleve property, Phys. Rev. A., 25 (1982), 1257-1264.
[7]	R. C. Churchill, D. L. Rod and M. F. Singer, Group-theoretic obstructions to integrability, Ergod. Th & Dynam. Sys. (1), 5 (1995), 15-48.
[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).
[9]	B. Dwork, Differential operators with nilponent $p$-curvature, Amer. J. Math., 112 (1990), 749-786. doi: 10.2307/2374806.
[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), 191-192. doi: 10.1007/BF00692087.
[11]	R. Fridberg, T. D. Lee and R. Padjen, Class of scalar-field solutions in three space dimensions, Phys. Rev. D., 13 (1976), 2739-2761.
[12]	G. H. Halphen, Traité des fonctions elliptiques VOl. I, II, Gauthier-Villars, Paris, (1888).
[13]	J. Hietarinta, Direct methods for the search of the second invariant, Phys. Rep., 147 (1987), 87-154. doi: 10.1016/0370-1573(87)90089-5.
[14]	S. Kasperczuk, Integrability of the Yang-Mills Hamiltonian system, Celest. Mech. Dynam. Astr., 58 (1994), 387-391.
[15]	W. L. Li and S. Y. Shi, Non-integrability of Hénon-Heiles System, Celest. Mech. Dynam. Astr., 109 (2010), 1-12.
[16]	A. J. Maciejewski, M. Przybylska, T. Stachowiak and M. Szydlowski, Global integrability of cosmological scalar fields, J. Phys. A., 41 (2008), 26 pp. 465101.
[17]	A. J. Maciejewski, M. Przybylska and J. A. Weil, Non-integrability of the generalized spring-pendulum problem, J. Phys. A., 37 (2004), 2579-2597.
[18]	A. J. Maciejewski and M. Przybylska, Darboux points and integrability of Hamiltonian systems with homogeneous polynomial potential, J. Math. Phys., 46 (2005), 062901.
[19]	S. V. Manakov, On the theory of two-dimensional stationary self-focusing of electromagnetic waves, Soviet Phys. JETP., 38 (1974), 248-253.
[20]	J. J. Morales-Ruiz, "Técnicas Algebraicas Para el Estudio de la Integrabilidad de Sistemas Hamiltonianos," Ph.D. Thesis, University of Barcelona, 1989.
[21]	J. J. Morales-Ruiz and C. Simó, Picard-Vessiot theory and Ziglin's theory, J. Differential Equations, 107 (1994), 140-162.
[22]	J. J. Morales-Ruiz, "Differential Galois Theory and Non-Integrability of Hamiltonian Systems," Birkhäuser Verlag, Basel, 1999.
[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), 111-135.
[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), 1237-1256.
[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), 845-884.
[26]	J. J. Morales-Ruiz and S. Simon, On the meromorphic non-integrability of some $N$-body problems, Discrete Contin. Dyn. Syst., 24 (2009), 1225-1273.
[27]	E. G. C. Poole, "Introduction to the Theory of Linear Differential Equations," Oxford Univ. Press, London, 1936.
[28]	R. Rajaraman and E. J. Weinberg, Internal symmetry and the semi-classical method in quantum field theory, Phys. Rev. D., 11 (1975), 2950-2966.
[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.
[30]	P. Vanhaecke, A special case of the Garnier system, (1,4)-polarised Abelian surfaces and their moduli, Compositio Math., 29 (1994), 157-203. doi: 10.1016/0165-0270(94)90123-6.
[31]	E. T. Whittaker and E. T. Watson, "A Course of Modern Analysis," Cambrige Univ. Press, Cambrige, 1969.
[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), 39-44.
[33]	S. L. Ziglin, Branching of solutions and non-existence of first integrals in Hamiltonian mechanics I, II, Funct. Anal. Appl., 16 (1983), 181-189; 6-17.

show all references

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), 299-314.
[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), 5-56.
[3]	F. Baldassarri, On Algebraic solution of Lamé's differential equation, J. Differential Equations, 41 (1981), 44-58.
[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), 263-278.
[5]	D. Boucher and J. A. Weil, About nonintegrability in the Friedmann-Robertson-Walker cosmological model, Brazilian Journal of Physics, 37 (2007), 398-405. doi: 10.1007/s10765-007-0152-8.
[6]	T. Bountis, H. Segur and F. Vivaldi, Integrable Hamiltonian systems and the Painleve property, Phys. Rev. A., 25 (1982), 1257-1264.
[7]	R. C. Churchill, D. L. Rod and M. F. Singer, Group-theoretic obstructions to integrability, Ergod. Th & Dynam. Sys. (1), 5 (1995), 15-48.
[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).
[9]	B. Dwork, Differential operators with nilponent $p$-curvature, Amer. J. Math., 112 (1990), 749-786. doi: 10.2307/2374806.
[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), 191-192. doi: 10.1007/BF00692087.
[11]	R. Fridberg, T. D. Lee and R. Padjen, Class of scalar-field solutions in three space dimensions, Phys. Rev. D., 13 (1976), 2739-2761.
[12]	G. H. Halphen, Traité des fonctions elliptiques VOl. I, II, Gauthier-Villars, Paris, (1888).
[13]	J. Hietarinta, Direct methods for the search of the second invariant, Phys. Rep., 147 (1987), 87-154. doi: 10.1016/0370-1573(87)90089-5.
[14]	S. Kasperczuk, Integrability of the Yang-Mills Hamiltonian system, Celest. Mech. Dynam. Astr., 58 (1994), 387-391.
[15]	W. L. Li and S. Y. Shi, Non-integrability of Hénon-Heiles System, Celest. Mech. Dynam. Astr., 109 (2010), 1-12.
[16]	A. J. Maciejewski, M. Przybylska, T. Stachowiak and M. Szydlowski, Global integrability of cosmological scalar fields, J. Phys. A., 41 (2008), 26 pp. 465101.
[17]	A. J. Maciejewski, M. Przybylska and J. A. Weil, Non-integrability of the generalized spring-pendulum problem, J. Phys. A., 37 (2004), 2579-2597.
[18]	A. J. Maciejewski and M. Przybylska, Darboux points and integrability of Hamiltonian systems with homogeneous polynomial potential, J. Math. Phys., 46 (2005), 062901.
[19]	S. V. Manakov, On the theory of two-dimensional stationary self-focusing of electromagnetic waves, Soviet Phys. JETP., 38 (1974), 248-253.
[20]	J. J. Morales-Ruiz, "Técnicas Algebraicas Para el Estudio de la Integrabilidad de Sistemas Hamiltonianos," Ph.D. Thesis, University of Barcelona, 1989.
[21]	J. J. Morales-Ruiz and C. Simó, Picard-Vessiot theory and Ziglin's theory, J. Differential Equations, 107 (1994), 140-162.
[22]	J. J. Morales-Ruiz, "Differential Galois Theory and Non-Integrability of Hamiltonian Systems," Birkhäuser Verlag, Basel, 1999.
[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), 111-135.
[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), 1237-1256.
[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), 845-884.
[26]	J. J. Morales-Ruiz and S. Simon, On the meromorphic non-integrability of some $N$-body problems, Discrete Contin. Dyn. Syst., 24 (2009), 1225-1273.
[27]	E. G. C. Poole, "Introduction to the Theory of Linear Differential Equations," Oxford Univ. Press, London, 1936.
[28]	R. Rajaraman and E. J. Weinberg, Internal symmetry and the semi-classical method in quantum field theory, Phys. Rev. D., 11 (1975), 2950-2966.
[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.
[30]	P. Vanhaecke, A special case of the Garnier system, (1,4)-polarised Abelian surfaces and their moduli, Compositio Math., 29 (1994), 157-203. doi: 10.1016/0165-0270(94)90123-6.
[31]	E. T. Whittaker and E. T. Watson, "A Course of Modern Analysis," Cambrige Univ. Press, Cambrige, 1969.
[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), 39-44.
[33]	S. L. Ziglin, Branching of solutions and non-existence of first integrals in Hamiltonian mechanics I, II, Funct. Anal. Appl., 16 (1983), 181-189; 6-17.

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