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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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