Periodic orbits and nonintegrability of generalized classical Yang–Mills Hamiltonian systems

Jiménez-Lara, Lidia; Llibre, Jaume

doi:10.1063/1.3559145

Cited by 20 publications

(23 citation statements)

References 29 publications

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“…Similar results to Theorems 1 and 2 were obtained by the authors for the Henon-Heiles [9] and YangMills [8] Hamiltonian systems with 2 parameters using the averaging theory of second and first order, respectively. Besides, as it is established in [9], the averaging method for finding isolated periodic orbits is an useful and relatively simple tool in order to find necessary conditions for showing the non-integrability of a Hamiltonian system.…”

Section: Introduction and Statement Of The Main Resultssupporting

confidence: 74%

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The cored and logarithm galactic potentials: Periodic orbits and integrability

Jiménez-Lara

Llibre

2012

Journal of Mathematical Physics

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Abstract. We apply the averaging theory of first order to study analytically families of periodic orbits for the cored and logarithmic Hamiltoniansandq 2 , which are relevant in the study of the galactic dynamic. We first show, after introducing a scale transformation in the coordinates and momenta with a parameter ε, that both systems give essentially the same set of equations of motion up to first order in ε. Then the conditions for finding families of periodic orbits, using the averaging theory up to firs order in ε, apply equally for both systems in every energy level H = h > 0 with H either H C or H L . We prove the existence of two periodic orbits if q is irrational, for ε small enough, and we give an analytic approximation for the initial conditions of these periodic orbits. Finally, the previous periodic orbits provide information about the non-integrability of the cored and the logarithmic Hamiltonian systems.

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Section: Introduction and Statement Of The Main Resultssupporting

confidence: 74%

“…This section follows essentially from section 4 of the reference [8]. We summarize some facts on the Liouville-Arnold integrability of the Hamiltonian systems, and on the theory of the periodic orbits of the differential equations.…”

Section: Proof Of Theoremmentioning

confidence: 99%

The cored and logarithm galactic potentials: Periodic orbits and integrability

Jiménez-Lara

Llibre

2012

Journal of Mathematical Physics

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“…Similar studies to the one done here for the cosmological coupled scalar field (5) about periodic orbits and their nonintegrability have been done for Yang-Mills, the Hénon-Heiles and the Armbruster-Guckenheimer-Kim Hamiltonians, see [10], [11] and [12], respectively.…”

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confidence: 74%

“…This method allows to find periodic orbits of our cosmological model (5), up to first order in ε, at any non-zero Hamiltonian level as a function of the parameters λ, Λ and m. Roughly speaking, this method reduces the problem of finding periodic solutions of some differential system to the one of finding zeros of some convenient finite dimensional function. In [10] and…”

Section: Introduction and Statements Of Main Resultsmentioning

confidence: 99%

Periodic orbits and non-integrability in a cosmological scalar field

Llibre

Vidal

2012

Journal of Mathematical Physics

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Abstract. We apply the averaging theory of first order to study the periodic orbits of Hamiltonian systems describing an universe filled with a scalar field which possesses three parameters. The main results are the following.First, we provide sufficient conditions on the parameters of these cosmological model, which guarantee that at any positive or negative Hamiltonian level, the Hamiltonian system has periodic orbits, the number of such periodic orbits and their stability change with the values of the parameters. These periodic orbits live in the whole phase space in a continuous family of periodic orbits parameterized by the Hamiltonian level.Second, under convenient assumptions we show the non-integrability of these cosmological systems in the sense of Liouville-Arnol'd, proving that cannot exist any second first integral of class C 1 .It is important to mention that the tools (i.e the averaging theory for studying the existence of periodic orbits and their kind of stability, and the multipliers of these periodic orbits for studying the integrability of the Hamiltonian system) used here for proving our results on the cosmological scalar field, can be applied to Hamiltonian systems with an arbitrary number of degrees of freedom.

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“…This method allows to find periodic orbits of our Hamiltonian system (3), up to first order in ε, at any Hamiltonian level H = h > 0 as a function of the parameter h. Roughly speaking, this method reduces the problem of finding periodic solutions of some differential system to the one of finding zeros of some convenient finite dimensional function. In [7,8] the application of this technique has been considered in order to obtain periodic solutions of some well known Hamiltonian systems.…”

Section: Introduction and Statements Of Main Resultsmentioning

confidence: 99%