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Mondéjar, F.; Vigueras, A.

doi:10.1023/a:1008375820146

Cited by 14 publications

(3 citation statements)

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“…A new interest in the study of configurations of relative equilibria by the use of differential geometry methods instead of more classical ones has appeared in recent years. See for instance, Wang et al [8] who treat the problem of a rigid body in a central Newtonian field or Maciejewski [5] for the problem of two rigid bodies under mutual Newtonian attraction.…”

Section: Introductionmentioning

confidence: 99%

“…A gyrostat is a mechanical system S composed by a rigid body S and other bodies S deformable or rigid, connected in such a way that their relative motion with respect to its rigid part does not change the distribution of masses of the total system (see Leimanis [4]). Results of papers [5] and [8] are generalized in Mondéjar et al [3] to the case of two gyrostats under mutual Newtonian attraction.…”

Section: Introductionmentioning

confidence: 99%

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Lagrangian relative equilibria for a gyrostat in the three-body problem: bifurcations and stability

Guirao¹,

Vera²

2010

J. Phys. A: Math. Theor.

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In this paper we consider the non-canonical Hamiltonian dynamics of a gyrostat in the frame of the three-body problem. Using geometric/mechanic methods we study the approximate dynamics of the truncated Legendre series representation of the potential of an arbitrary order. Working in the reduced problem, we study the existence of relative equilibria that we refer to as Lagrange type following the analogy with the standard techniques. We provide necessary and sufficient conditions for the linear stability of Lagrangian relative equilibria if the gyrostat morphology form is close to a sphere. Thus, we generalize the classical results on equilibria of the three-body problem and many results on them obtained by the classic approach for the case of rigid bodies.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Lagrangian relative equilibria for a gyrostat in the three-body problem: bifurcations and stability

Guirao¹,

Vera²

2010

J. Phys. A: Math. Theor.

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“…These papers have been generalized to the case of a gyrostat by Mondéjar, et al [12] to the case of two gyrostats in mutual Newtonian attraction.…”

Section: Introductionmentioning

confidence: 99%

Lagrangian relative equilibria for a gyrostat in the three-body problem

López

2009

Open Physics

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Abstract:In this paper we consider the noncanonical Hamiltonian dynamics of a gyrostat in the three-body problem. By means of geometric mechanics methods, we study the approximate Poisson dynamics that arise when we develop the potential of the system in Legendre series and truncate this to an arbitrary order . After reduction of the dynamics by means of the two symmetries of the system, we consider the existence and number of equilibria which we denominate of Lagrangian type, in analogy with classic results on the topic. Necessary and sufficient conditions are established for their existence in an approximate dynamics of order , and explicit expressions for these equilibria are given, this being useful for the subsequent study of their stability. The number of Lagrangian equilibria is thoroughly studied in approximate dynamics of orders zero and one. The main result of this work indicates that the number of Lagrangian equilibria in an approximate dynamics of order for ≥ 1 is independent of the order of truncation of the potential, if the gyrostat S 0 is almost spherical. In relation to the stability of these equilibria, necessary and sufficient conditions are given for linear stability of Lagrangian equilibria when the gyrostat is almost spherical. In this way, we generalize the classical results on equilibria of the three-body problem and many results provided by other authors using more classical techniques for the case of rigid bodies. 45.20.dc, 45.20.Jj, 45.50.Pk, 47.10.Df PACS (2008): Keywords:hamiltonian system • three-body problem • gyrostat • lagrangian equilibria • stability © Versita Warsaw and Springer-Verlag Berlin Heidelberg.

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