Problem In Spaces Of Constant Curvature Research Articles

The restricted two-body problem in constant curvature spaces

We perform the bifurcation analysis of the Kepler problem on $S^3$ and $L^3$. An analogue of the Delaunay variables is introduced. We investigate the motion of a point mass in the field of the Newtonian center moving along a geodesic on $S^2$ and $L^2$ (the restricted two-body problem). When the curvature is small, the pericenter shift is computed using the perturbation theory. We also present the results of the numerical analysis based on the analogy with the motion of rigid body.

Nonintegrability of the two-body problem in constant curvature spaces

We consider the reduced two-body problem with the Newton and the oscillator potentials on the sphere S2 and the hyperbolic plane H2. For both types of interaction we prove the nonexistence of an additional meromorphic integral for the complexified dynamic systems.

Comment on “Central potentials on spaces of constant curvature: The Kepler problem on the two-dimensional sphere S2 and the hyperbolic plane H2” [J. Math. Phys. 46, 052702 (2005)

Cariñena, Rañada, and Santander consider a classical Kepler problem in constant curvature spaces and related theory of conics in these spaces. Here we point out that earlier results exist in the literature. Some of these results date back to the nineteenth and the beginning of the twentieth century.

Kepler's problem in constant curvature spaces

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 are closed in two cases: when the potential satisfies Iaplace-Beltrami equation and can be regarded as an analogue of the potential of the gravitational interaction, and in the case when the potential is the generalization of the potential of an elastic spring. Also the full integrability of the generalized two-centre problem on a constant curvature surface is discovered and it is shown that integrability remains even if elastic “forces” are added.