Borisov, A.V. and GarcíaNaranjo, L.C. and Mamaev, I.S. and Montaldi, James (2018) Reduction and relative equilibria for the 2body problem in spaces of constant curvature. Celestial Mechanics and Dynamical Astronomy, 130. ISSN 09232958
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Abstract
We consider the twobody 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. Rightangled 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 5dimensional phase space and possess one Casimir function.
Item Type:  Article 

Subjects:  MSC 2010, the AMS's Mathematics Subject Classification > 70 Mechanics of particles and systems 
Depositing User:  Dr James Montaldi 
Date Deposited:  18 Feb 2018 08:35 
Last Modified:  27 Sep 2018 16:13 
URI:  http://eprints.maths.manchester.ac.uk/id/eprint/2623 
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Reduction and relative equilibria for the 2body problem in spaces of constant curvature. (deposited 12 Aug 2017)
 Reduction and relative equilibria for the 2body problem in spaces of constant curvature. (deposited 18 Feb 2018 08:35) [Currently Displayed]
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