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The restricted two-body problem in constant curvature spaces. (English) Zbl 1116.70015
Summary: We 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 mass in the field of a Newtonian center moving along a geodesic on $$\mathbb{S}^{2}$$ and $$\mathbb{H}^{2}$$ (the restricted two-body problem). For the case of a small curvature, the pericenter shift is computed using the perturbation theory. We also present the results of numerical analysis based on an analogy with the motion of a rigid body.

##### MSC:
 70F05 Two-body problems 37J20 Bifurcation problems for finite-dimensional Hamiltonian and Lagrangian systems 37N05 Dynamical systems in classical and celestial mechanics 70G45 Differential geometric methods (tensors, connections, symplectic, Poisson, contact, Riemannian, nonholonomic, etc.) for problems in mechanics
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