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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
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.  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), 5-56.  Google Scholar

[3]

F. Baldassarri, On Algebraic solution of Lamé's differential equation, J. Differential Equations, 41 (1981), 44-58.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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T. Bountis, H. Segur and F. Vivaldi, Integrable Hamiltonian systems and the Painleve property, Phys. Rev. A., 25 (1982), 1257-1264.  Google Scholar

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R. C. Churchill, D. L. Rod and M. F. Singer, Group-theoretic obstructions to integrability, Ergod. Th & Dynam. Sys. (1), 5 (1995), 15-48.  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

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B. Dwork, Differential operators with nilponent $p$-curvature, Amer. J. Math., 112 (1990), 749-786. doi: 10.2307/2374806.  Google Scholar

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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.  Google Scholar

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R. Fridberg, T. D. Lee and R. Padjen, Class of scalar-field solutions in three space dimensions, Phys. Rev. D., 13 (1976), 2739-2761.  Google Scholar

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G. H. Halphen, Traité des fonctions elliptiques VOl. I, II, Gauthier-Villars, Paris, (1888). Google Scholar

[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.  Google Scholar

[14]

S. Kasperczuk, Integrability of the Yang-Mills Hamiltonian system, Celest. Mech. Dynam. Astr., 58 (1994), 387-391.  Google Scholar

[15]

W. L. Li and S. Y. Shi, Non-integrability of Hénon-Heiles System, Celest. Mech. Dynam. Astr., 109 (2010), 1-12.  Google Scholar

[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.  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), 2579-2597.  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), 062901.  Google Scholar

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S. V. Manakov, On the theory of two-dimensional stationary self-focusing of electromagnetic waves, Soviet Phys. JETP., 38 (1974), 248-253. Google Scholar

[20]

J. J. Morales-Ruiz, "Técnicas Algebraicas Para el Estudio de la Integrabilidad de Sistemas Hamiltonianos," Ph.D. Thesis, University of Barcelona, 1989. Google Scholar

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J. J. Morales-Ruiz and C. Simó, Picard-Vessiot theory and Ziglin's theory, J. Differential Equations, 107 (1994), 140-162.  Google Scholar

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J. J. Morales-Ruiz, "Differential Galois Theory and Non-Integrability of Hamiltonian Systems," Birkhäuser Verlag, Basel, 1999.  Google Scholar

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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.  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), 1237-1256.  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), 845-884.  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), 1225-1273.  Google Scholar

[27]

E. G. C. Poole, "Introduction to the Theory of Linear Differential Equations," Oxford Univ. Press, London, 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), 2950-2966. 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), 157-203. 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, Cambrige, 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), 39-44. 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), 181-189; 6-17. Google Scholar

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.  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), 5-56.  Google Scholar

[3]

F. Baldassarri, On Algebraic solution of Lamé's differential equation, J. Differential Equations, 41 (1981), 44-58.  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), 263-278.  Google Scholar

[5]

D. Boucher and J. A. Weil, About nonintegrability in the Friedmann-Robertson-Walker cosmological model, Brazilian Journal of Physics, 37 (2007), 398-405. 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), 1257-1264.  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), 15-48.  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), 749-786. 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), 191-192. 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), 2739-2761.  Google Scholar

[12]

G. H. Halphen, Traité des fonctions elliptiques VOl. I, II, Gauthier-Villars, Paris, (1888). Google Scholar

[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.  Google Scholar

[14]

S. Kasperczuk, Integrability of the Yang-Mills Hamiltonian system, Celest. Mech. Dynam. Astr., 58 (1994), 387-391.  Google Scholar

[15]

W. L. Li and S. Y. Shi, Non-integrability of Hénon-Heiles System, Celest. Mech. Dynam. Astr., 109 (2010), 1-12.  Google Scholar

[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.  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), 2579-2597.  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), 062901.  Google Scholar

[19]

S. V. Manakov, On the theory of two-dimensional stationary self-focusing of electromagnetic waves, Soviet Phys. JETP., 38 (1974), 248-253. Google Scholar

[20]

J. J. Morales-Ruiz, "Técnicas Algebraicas Para el Estudio de la Integrabilidad de Sistemas Hamiltonianos," Ph.D. Thesis, University of Barcelona, 1989. Google Scholar

[21]

J. J. Morales-Ruiz and C. Simó, Picard-Vessiot theory and Ziglin's theory, J. Differential Equations, 107 (1994), 140-162.  Google Scholar

[22]

J. J. Morales-Ruiz, "Differential Galois Theory and Non-Integrability of Hamiltonian Systems," Birkhäuser Verlag, Basel, 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), 111-135.  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), 1237-1256.  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), 845-884.  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), 1225-1273.  Google Scholar

[27]

E. G. C. Poole, "Introduction to the Theory of Linear Differential Equations," Oxford Univ. Press, London, 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), 2950-2966. 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), 157-203. 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, Cambrige, 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), 39-44. 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), 181-189; 6-17. Google Scholar

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