On the Ziglin-Yoshida analysis for some classes of homogeneous hamiltonian systems

Almeida, Marcos Antônio de; Moreira, I.; Santos, F. C.

doi:10.1590/s0103-97331998000400022

Cited by 10 publications

(13 citation statements)

References 3 publications

(3 reference statements)

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“…The non-integrability of the reduced (by the S 1 symmetry) SP was established by Almeida et al [2] and Saenz, Kummer [27]. In the latter paper the implications, of the non-integrability of the complexified reduced SP, to the original (real-dynamics) SP are also considered.…”

Section: Introductionmentioning

confidence: 89%

See 1 more Smart Citation

Non-integrability of the Anisotropic Stormer Problem and the Isosceles Three-Body Problem

Νομικός

Papageorgiou

2009

Physica D: Nonlinear Phenomena

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Section: Introductionmentioning

confidence: 89%

“…If system (2), with V (q) a homogeneous function of degree k ∈ Z, is integrable, then (A) for |k| = 1 or |k| ≥ 6 each λ i , i = 1, 2, . .…”

Section: Theoretical Preliminariesmentioning

confidence: 99%

Non-integrability of the Anisotropic Stormer Problem and the Isosceles Three-Body Problem

Νομικός

Papageorgiou

2009

Physica D: Nonlinear Phenomena

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“…The Contopoulos Hamiltonian describes the perturbed the term x 2 y 2 characterize the Yang-Mills potential, which arises in connection with the classical Yang-Mills field with gauge group SU(2) for a homogeneous two-component field [18]. Quartic homogeneous potentials (without quadratic terms) have been studied by several authors, see for instance [2,5,16], and it is well known that the Hamiltonian H Y M with b = 0 is non-integrable and strongly chaotic. Generalizations of the mechanical Yang-Mills Hamiltonian, with three up to five quartic terms, have been considered in [9,15,23,25].…”

Section: Introductionmentioning

confidence: 99%

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

Jiménez-Lara

Llibre²

2011

Journal of Mathematical Physics

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Abstract. The averaging theory of first order is applied to study a generalized Yang-Mills system with two parameters. Two main results are proved. First, we provide sufficient conditions on the two parameters of the generalized system to guarantee the existence of continuous families of isolated periodic orbits parameterized by the energy, and these families are given up to first order in a small parameter. Second, we prove that for the non-integrable classical Yang-Mills Hamiltonian systems, in the sense of Liouville-Arnold, which have the isolated periodic orbits found with averaging theory, can not exist any second first integral of class C 1 . This is important because most of the results about integrability deals with analytic or meromorphic integrals of motion.

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“…For µ > 3, [9] proved that SAM without massive pulleys is not integrable, contrary to what was speculated by [27]. The belonging of µ = M/m to a special set of parameters {µ p : p ∈ Z} was established as a necessary condition for integrability; this result was proven independently in [9], [35] and [2], and is proven in Remark 7.1(2) of the present paper as well. Moreover, unbounded trajectories (µ ≤ 1) have been studied via energetic considerations [31]; [20] identified and classified all periodic trajectories in the pulley-less SAM for µ = 3.…”

Section: Introductionmentioning

confidence: 49%

Swinging Atwood Machine: Experimental and numerical results, and a theoretical study

Pujol¹,

Pérez²,

Ramis³

et al. 2010

Physica D: Nonlinear Phenomena

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A Swinging Atwood Machine (SAM ) is built and some experimental results concerning its dynamic behaviour are presented. Experiments clearly show that pulleys play a role in the motion of the pendulum, since they can rotate and have non-negligible radii and masses. Equations of motion must therefore take into account the inertial momentum of the pulleys, as well as the winding of the rope around them. Their influence is compared to previous studies. A preliminary discussion of the role of dissipation is included. The theoretical behaviour of the system with pulleys is illustrated numerically, and the relevance of different parameters is highlighted. Finally, the integrability of the dynamic system is studied, the main result being that the Machine with pulleys is non-integrable. The status of the results on integrability of the pulley-less Machine is also recalled.Key words: Swinging Atwood's Machine (SAM ), Chaotic system, Nonlinear dynamics, Integrability, Experimental SAM apparatus. * Corresponding author Email addresses: olivier.pujol@loa.univ-lille1.fr (O Pujol), perez@ast.obs-mip.fr (J P Pérez), ramis.jean-pierre@wanadoo.fr (J P Ramis), carles@maia.ub.es (C Simó), Sergi.Simon@port.ac.uk (S Simon), Jacques-Arthur.Weil@unilim.fr (J A Weil)

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On the Ziglin-Yoshida analysis for some classes of homogeneous hamiltonian systems

Cited by 10 publications

References 3 publications

Non-integrability of the Anisotropic Stormer Problem and the Isosceles Three-Body Problem

Non-integrability of the Anisotropic Stormer Problem and the Isosceles Three-Body Problem

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

Swinging Atwood Machine: Experimental and numerical results, and a theoretical study

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