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

@article{Borisov2018ReductionAR, title={Reduction and relative equilibria for the two-body problem on spaces of constant curvature}, author={Alexey Vladimirovich Borisov and Luis C. Garc{\'i}a-Naranjo and Ivan S. Mamaev and James Montaldi}, journal={Celestial Mechanics and Dynamical Astronomy}, year={2018}, volume={130}, pages={1-36} }

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 $$q>0$$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…

## 14 Citations

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## References

SHOWING 1-10 OF 36 REFERENCES

Central potentials on spaces of constant curvature: The Kepler problem on the two-dimensional sphere S2 and the hyperbolic plane H2

- Mathematics, Physics
- 2005

The Kepler problem is a dynamical system that is well defined not only on the Euclidean plane but also on the sphere and on the hyperbolic plane. First, the theory of central potentials on spaces of…

Point vortices on the hyperbolic plane

- Physics
- 2014

We investigate the dynamical system of point vortices on the hyperboloid. This system has non-compact symmetry SL(2, R) and a coadjoint equivariant momentum map. The relative equilibrium conditions…

An intrinsic approach in the curved n-body problem. The positive curvature case

- Mathematics
- 2012

We consider the gravitational motion of n point particles with masses m1,m2, . . . ,mn > 0 on surfaces of constant positive Gaussian curvature. Using stereographic projection, we express the…

Two-body problem on spaces of constant curvature

- Mathematics
- 2005

The two-body problem with a central interaction on simply connected constant curvature spaces of an arbitrary dimension is considered. The explicit expression for the quantum two-body Hamiltonian via…

The restricted two-body problem in constant curvature spaces

- Mathematics, Physics
- 2005

AbstractWe perform the bifurcation analysis of the Kepler problem on
$$\mathbb{S}^{3}$$ and
$$\mathbb{H}^{3}$$. An analog of the Delaunay variables is introduced. We investigate the motion of a point…

Kepler's problem in constant curvature spaces

- Mathematics
- 1992

In this article the generalization of the motion of a particle in a central field to the case of a constant curvature space is investigated. We found out that orbits on a constant curvature surface…

Rigid body dynamics in non-Euclidean spaces

- Mathematics
- 2016

In this paper, we focus on the study of two-dimensional plate dynamics on the Lobachevskii plane L2. First of all, we consider the free motion of such a plate, which is a pseudospherical analog of…

The spatial problem of 2 bodies on a sphere. Reduction and stochasticity

- Mathematics
- 2016

In this paper, we consider in detail the 2-body problem in spaces of constant positive curvature S2 and S3. We perform a reduction (analogous to that in rigid body dynamics) after which the problem…

Topology and stability of integrable systems

- Mathematics
- 2010

In this paper a general topological approach is proposed for the study of stability of periodic solutions of integrable dynamical systems with two degrees of freedom. The methods developed are…