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

Abstract

We consider the two-body problem on surfaces of constant nonzero 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 \(\pi /2\). 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 \(\pi /2\), 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 five-dimensional phase space and possess one Casimir function.

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Notes

  1. 1.

    We mention only the paper (Bolsinov et al. 2012) on the Conley index where new isosceles vortex configurations were found and their stability was established using topological methods.

  2. 2.

    This assumption is done without loss of generality since one may eliminate these quantities from the equations of motion (2.8) by rescaling time \(t\rightarrow \frac{\sqrt{\varkappa } t}{\mu _1}\), and the momenta \(p\rightarrow \frac{p}{\sqrt{\varkappa }} , \; {\varvec{m}}\rightarrow \frac{ \varvec{m}}{\sqrt{\varkappa }}\), where \(\varkappa =\mu _1\sqrt{G\mu _2}\).

  3. 3.

    An equivalent proof of Theorem 3.2 was given before in García-Naranjo et al. (2016) by working on a symplectic slice in the unreduced system, since the reduced equations of motion were not known at that time. Although equivalent, the approach that we follow in this paper is elementary.

  4. 4.

    In fact \((\alpha ,q,z,p)\) as defined are Darboux coordinates that generalize the Andoyer variables on \(T^*SO(3)\).

  5. 5.

    As before, \((\alpha ,q,z,p)\) are Darboux coordinates that generalize the Andoyer variables on \(T^*SO(3)\).

  6. 6.

    An alternative proof is given in (2018) by working on a symplectic slice of the unreduced system.

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Acknowledgements

We are thankful to both reviewers and the associate editor for their remarks and criticisms which led to an improvement of our paper. We are grateful to Miguel Rodríguez-Olmos for discussing his preliminary results of (2018) with us. The authors express their gratitude to B. S. Bardin and I. A. Bizyaev for fruitful discussions and useful comments. The research contribution of LGN and JM was made possible by a Newton Advanced Fellowship from the Royal Society, Ref: NA140017. The work of AVB and ISM is supported by the Russian Foundation for Basic Research (Project No. 17-01-00846-a). The research of AVB was also carried out within the framework of the state assignment of the Ministry of Education and Science of Russia.

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Borisov, A.V., García-Naranjo, L.C., Mamaev, I.S. et al. Reduction and relative equilibria for the two-body problem on spaces of constant curvature. Celest Mech Dyn Astr 130, 43 (2018). https://doi.org/10.1007/s10569-018-9835-7

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Keywords

  • Reduction
  • Relative equilibria
  • Hamiltonian systems
  • Stability
  • Two-body problem
  • Energy–momentum bifurcation diagram