Classification and stability of relative equilibria for the two-body problem in the hyperbolic space of dimension 2

García-Naranjo, Luis C.; Marrero, Juan Carlos; Pérez‐Chavela, Ernesto; Rodriguez-Olmos, Miguel

doi:10.1016/j.jde.2015.12.044

Cited by 31 publications

(39 citation statements)

References 28 publications

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“…After finding the reduced equations, in the final two sections we proceed to classify the RE of the problem by finding all the equilibria of the reduced equations. In this way we recover the results of [1,12,13] in a systematic and elementary fashion. Moreover, with this approach, we are able to conveniently analyse their stability.…”

Section: Classification and Stability Of Relative Equilibriamentioning

confidence: 69%

“…This classification becomes very transparent in our treatment: hyperbolic RE correspond to equilibria of the reduced Hamiltonian system restricted to negative values of the Casimir function C , whereas elliptic RE are those for positive values of C (see Section 2.3 below). It is also known that parabolic transformations do not give rise to RE of the problem [12,13]. In our treatment, this corresponds to the absence of equilibria of the reduced system when C = 0.…”

Section: The Case Of Negative Curvaturementioning

confidence: 87%

“…The (nonlinear) stability properties of the RE described above was first established in [13] by working on a symplectic slice for the unreduced system since the reduced equations were not available (in a journal in English!). With the reduced system at hand, we are able to recover these results in elementary terms by directly analysing the signature of the Hessian of the reduced Hamiltonian at the RE.…”

Section: The Case Of Negative Curvaturementioning

confidence: 99%

“…13) that will be relevant in what follows. To obtain the simplified expression on the right, one needs to use (4.10) and then (4.6a) to eliminate µ.…”

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confidence: 99%

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Reduction and relative equilibria for the two-body problem on spaces of constant curvature

Borisov

García-Naranjo

Mamaev

et al. 2018

Celest Mech Dyn Astr

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We consider the two-body problem on surfaces of constant non-zero curvature and classify the relative equilibria and their stability. On the hyperbolic plane, for each q > 0 we show there are two relative equilibria where the masses are separated by a distance q. One of these is geometrically of elliptic type and the other of hyperbolic type. The hyperbolic ones are always unstable, while the elliptic ones are stable when sufficiently close, but unstable when far apart.On the sphere of positive curvature, if the masses are different, there is a unique relative equilibrium (RE) for every angular separation except π/2. When the angle is acute, the RE is elliptic, and when it is obtuse the RE can be either elliptic or linearly unstable. We show using a KAM argument that the acute ones are almost always nonlinearly stable. If the masses are equal there are two families of relative equilibria: one where the masses are at equal angles with the axis of rotation ('isosceles RE') and the other when the two masses subtend a right angle at the centre of the sphere. The isosceles RE are elliptic if the angle subtended by the particles is acute and is unstable if it is obtuse. At π/2, the two families meet and a pitchfork bifurcation takes place. Right-angled RE are elliptic away from the bifurcation point.In each of the two geometric settings, we use a global reduction to eliminate the group of symmetries and analyse the resulting reduced equations which live on a 5-dimensional phase space and possess one Casimir function.

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Section: Classification and Stability Of Relative Equilibriamentioning

confidence: 69%

Section: The Case Of Negative Curvaturementioning

confidence: 87%

Section: The Case Of Negative Curvaturementioning

confidence: 99%

“…13) that will be relevant in what follows. To obtain the simplified expression on the right, one needs to use (4.10) and then (4.6a) to eliminate µ.…”

mentioning

confidence: 99%

See 2 more Smart Citations

Reduction and relative equilibria for the two-body problem on spaces of constant curvature

Borisov

García-Naranjo

Mamaev

et al. 2018

Celest Mech Dyn Astr

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“…Unlike in Euclidean space, the equations describing them are not equivalent, since the latter system is not integrable, [20]. More recently, the problem was generalized to any number N of bodies, leading to works such as [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [18], [19], [20], [21], [22], [23], and [24]. In the light of Hubble's law, [16], a non-flat universe (i.e.…”

Section: Introductionmentioning

confidence: 99%

The N-body problem in spaces with uniformly varying curvature

Eric

Diacu

Zhu

2017

Journal of Mathematical Physics

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We generalize the curved N -body problem to spheres and hyperbolic spheres whose curvature κ varies in time. Unlike in the particular case when the curvature is constant, the equations of motion are non-autonomous. We first briefly consider the analogue of the Kepler problem and then investigate the homographic orbits for any number of bodies, proving the existence of several such classes of solutions on spheres. Allowing the curvature to vary in time offers some insight into the effect of an expanding universe, in the context the curved N -body problem, when κ satisfies Hubble's law. The study of these equations also opens the possibility of finding new connections between classical mechanics and general relativity.

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Central Configurations of the Curved N-Body Problem

2018

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We consider the N -body problem of celestial mechanics in spaces of nonzero constant curvature. Using the concept of locked inertia tensor, we compute the moment of inertia for systems moving on spheres and hyperbolic spheres and show that we can recover the classical definition in the Euclidean case. After proving some criteria for the existence of relative equilibria, we find a natural way to define the concept of central configuration in curved spaces using the moment of inertia, and show that our definition is formally similar to the one that governs the classical problem. The existence criteria we develop for central configurations help us provide several examples and prove that, for any given point masses on spheres and hyperbolic spheres, central configurations always exist. We end our paper with results concerning the number of central configurations that lie on the same geodesic, thus extending the celebrated theorem of Moulton to hyperbolic spheres and pointing out that it has no straightforward generalization to spheres, where the count gets complicated even in the case N = 2.

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Classification and stability of relative equilibria for the two-body problem in the hyperbolic space of dimension 2

Abstract: Abstract. We classify and analyze the stability of all relative equilibria for the two-body problem in the hyperbolic space of dimension 2 and we formulate our results in terms of the intrinsic Riemannian data of the problem.

Cited by 31 publications

References 28 publications

Reduction and relative equilibria for the two-body problem on spaces of constant curvature

Reduction and relative equilibria for the two-body problem on spaces of constant curvature

The N-body problem in spaces with uniformly varying curvature

Central Configurations of the Curved N-Body Problem

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