Non-integrability of generalized Yang-Mills Hamiltonian system

Shaoyun Shi; Wenlei Li

doi:10.3934/dcds.2013.33.1645

Article Contents

2013, Volume 33, Issue 4: 1645-1655. Doi: 10.3934/dcds.2013.33.1645

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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
2.
College of Mathematics, Jilin University, Changchun 130012

Received: October 31, 2011

Revised: April 30, 2012

Published: March 2013

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Abstract

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.

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Mathematics Subject Classification: Primary: 70H06, 34M15; Secondary: 34M03.

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