American Institute of Mathematical Sciences

Advanced
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 - A, 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.   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

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.   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

[1]

Regina Martínez, Carles Simó. Non-integrability of the degenerate cases of the Swinging Atwood's Machine using higher order variational equations. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 1-24. doi: 10.3934/dcds.2011.29.1
[2]

Hartmut Pecher. Infinite energy solutions for the (3+1)-dimensional Yang-Mills equation in Lorenz gauge. Communications on Pure & Applied Analysis, 2019, 18 (2) : 663-688. doi: 10.3934/cpaa.2019033
[3]

Alberto Ibort, Amelia Spivak. Covariant Hamiltonian field theories on manifolds with boundary: Yang-Mills theories. Journal of Geometric Mechanics, 2017, 9 (1) : 47-82. doi: 10.3934/jgm.2017002
[4]

Andrew N. W. Hone, Michael V. Irle. On the non-integrability of the Popowicz peakon system. Conference Publications, 2009, 2009 (Special) : 359-366. doi: 10.3934/proc.2009.2009.359
[5]

Primitivo B. Acosta-Humánez, Martha Alvarez-Ramírez, David Blázquez-Sanz, Joaquín Delgado. Non-integrability criterium for normal variational equations around an integrable subsystem and an example: The Wilberforce spring-pendulum. Discrete & Continuous Dynamical Systems - A, 2013, 33 (3) : 965-986. doi: 10.3934/dcds.2013.33.965
[6]

Guillaume Duval, Andrzej J. Maciejewski. Integrability of Hamiltonian systems with homogeneous potentials of degrees $\pm 2$. An application of higher order variational equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4589-4615. doi: 10.3934/dcds.2014.34.4589
[7]

Primitivo Acosta-Humánez, David Blázquez-Sanz. Non-integrability of some hamiltonians with rational potentials. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 265-293. doi: 10.3934/dcdsb.2008.10.265
[8]

Juan J. Morales-Ruiz, Sergi Simon. On the meromorphic non-integrability of some $N$-body problems. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1225-1273. doi: 10.3934/dcds.2009.24.1225
[9]

Mitsuru Shibayama. Non-integrability of the collinear three-body problem. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 299-312. doi: 10.3934/dcds.2011.30.299
[10]

Delia Schiera. Existence and non-existence results for variational higher order elliptic systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 5145-5161. doi: 10.3934/dcds.2018227
[11]

Anthony Bloch, Leonardo Colombo, Fernando Jiménez. The variational discretization of the constrained higher-order Lagrange-Poincaré equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 309-344. doi: 10.3934/dcds.2019013
[12]

Guillaume Duval, Andrzej J. Maciejewski. Integrability of potentials of degree $k \neq \pm 2$. Second order variational equations between Kolchin solvability and Abelianity. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 1969-2009. doi: 10.3934/dcds.2015.35.1969
[13]

Sergi Simon. Linearised higher variational equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4827-4854. doi: 10.3934/dcds.2014.34.4827
[14]

Mitsuru Shibayama. Non-integrability criterion for homogeneous Hamiltonian systems via blowing-up technique of singularities. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3707-3719. doi: 10.3934/dcds.2015.35.3707
[15]

Xuewei Cui, Mei Yu. Non-existence of positive solutions for a higher order fractional equation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1379-1387. doi: 10.3934/dcds.2019059
[16]

Chiara Leone, Anna Verde, Giovanni Pisante. Higher integrability results for non smooth parabolic systems: The subquadratic case. Discrete & Continuous Dynamical Systems - B, 2009, 11 (1) : 177-190. doi: 10.3934/dcdsb.2009.11.177
[17]

David F. Parker. Higher-order shallow water equations and the Camassa-Holm equation. Discrete & Continuous Dynamical Systems - B, 2007, 7 (3) : 629-641. doi: 10.3934/dcdsb.2007.7.629
[18]

Min Zhu. On the higher-order b-family equation and Euler equations on the circle. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 3013-3024. doi: 10.3934/dcds.2014.34.3013
[19]

Eduardo Martínez. Higher-order variational calculus on Lie algebroids. Journal of Geometric Mechanics, 2015, 7 (1) : 81-108. doi: 10.3934/jgm.2015.7.81
[20]

Belhassen Dehman, Jean-Pierre Raymond. Exact controllability for the Lamé system. Mathematical Control & Related Fields, 2015, 5 (4) : 743-760. doi: 10.3934/mcrf.2015.5.743

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (22)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences