Phase portraits of the two-body problem with Manev potential

Llibre, Jaume; Teruel, Antonio E.; Valls, Clàudìa; Fuente, Alex de la

doi:10.1088/0305-4470/34/9/309

Cited by 17 publications

(22 citation statements)

References 8 publications

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“…Finally, let S n−1 be the sphere in R n , with n > 1 and Y the union of two open solid tori identifying point to point the points of two circles of each torus which cannot be contracted to a single point inside the corresponding torus (see [4] for details).…”

Section: Qualitative Study Of the Hamiltonian Flowmentioning

confidence: 99%

“…In order to do a qualitative study of the dynamics associated to the Hamiltonian system, in a similar way to [4], we consider the following sets:…”

Section: Introductionmentioning

confidence: 99%

“…The main tool for this study is the Liouville-Arnold theorem ( [1], [4]), applied to the momentum map (H , p θ ) : E × R −→ R 2 at regular values. A particular study for the sets I h , I k and I hk for critical values of the momentum map is made.…”

Section: Introductionmentioning

confidence: 99%

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The Motion of a Gyrostat in a Central Gravitational Field: Phase Portraits of an Integrable Case

Balsas¹,

Jiménez²,

Vera³

2008

JNMP

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In this paper we describe the Hamiltonian dynamics, in some invariant manifolds of the motion of a gyrostat in Newtonian interaction with a spherical rigid body. Considering a first integrable approximation of this roto-translatory problem, by means of Liouville-Arnold theorem and some specifics techniques, we obtained a complete topological classification of the phase flow associated to this system. The action-angle variables regions are obtained. These variables allow us to calculate the modified Keplerian elements of this problem useful to elaborate a perturbation theory. The results of this work have a direct application to the study of two body roto-translatory pro-blems where the rotation of one of them influences strongly in the orbital motion of the system. In particular, we can apply these results to binary asteroids.

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Section: Qualitative Study Of the Hamiltonian Flowmentioning

confidence: 99%

“…In order to do a qualitative study of the dynamics associated to the Hamiltonian system, in a similar way to [4], we consider the following sets:…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

The Motion of a Gyrostat in a Central Gravitational Field: Phase Portraits of an Integrable Case

Balsas¹,

Jiménez²,

Vera³

2008

JNMP

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show abstract

“…Finally, E = R + × S 1 × R 2 is the phase space. In order to do a qualitative study of the dynamics associated with the Hamiltonian system, similar to [6], we are going to consider the following sets:…”

Section: Introductionmentioning

confidence: 99%

“…This formation provides a good description of the phase space when ( ) ∈ R 2 and depends on the different values of α and β. The main tool for this study is the Liouville-Arnold theorem ( [2,6]), applied to the momentum map ( θ ) :…”

Section: Introductionmentioning

confidence: 99%

Qualitative analysis of the phase flow of an integrable approximation of a generalized roto-translatory problem

et al. 2009

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Abstract:In this paper, we consider an integrable approximation of the planar motion of a gyrostat in Newtonian interaction with a spherical rigid body. We then describe the Hamiltonian dynamics, in the fibers of constant total angular momentum vector of an invariant manifold of motion. Finally, using the Liouville-Arnold theorem and a particular analysis of the momentum map in its critical points, we obtain a complete topological classification of the different invariant sets of the phase flow of this problem. The results can be applied to study two-body roto-translatory problems where the rotation of one of them has a strong influence on the orbital motion of the system. 45.20.dc, 45.20.Jj, 45.40.Cc, 45.50.Pk PACS (2008):

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Linear Stability of the Lagrangian Triangle Solutions for Quasihomogeneous Potentials

Santoprete

2006

Celestial Mech Dyn Astr

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In this paper, we study the linear stability of the relative equilibria for homogeneous and quasihomogeneous potentials. First, in the case the potential is a homogeneous function of degree −a, we find that any relative equilibrium of the n-body problem with a > 2 is spectrally unstable. We also find a similar condition in the quasihomogeneous case. Then we consider the case of three bodies and we study the stability of the equilateral triangle relative equilibria. In the case of homogeneous potentials we recover the classical result obtained by Routh in a simpler way. In the case of quasihomogeneous potentials we find a generalization of Routh inequality and we show that, for certain values of the masses, the stability of the relative equilibria depends on the size of the configuration.

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Phase portraits of the two-body problem with Manev potential

Cited by 17 publications

References 8 publications

The Motion of a Gyrostat in a Central Gravitational Field: Phase Portraits of an Integrable Case

The Motion of a Gyrostat in a Central Gravitational Field: Phase Portraits of an Integrable Case

Qualitative analysis of the phase flow of an integrable approximation of a generalized roto-translatory problem

Linear Stability of the Lagrangian Triangle Solutions for Quasihomogeneous Potentials

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