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Nonintegrability and chaos in the anisotropic Manev problem. (English) Zbl 0996.70015

Summary: The anisotropic Manev problem, which lies at the intersection of classical, quantum, and relativity physics, describes the motion of two point masses in an anisotropic space under the influence of a Newtonian force law with a relativistic correction term. Using an extension of Poincaré-Melnikov method, we first prove that for weak anisotropy, chaos shows up on the zero-energy manifold. Then we put into evidence a class of isolated periodic orbits, and show that the system is nonintegrable. Finally, using the geodesic deviation approach, we prove the existence of a large nonchaotic set of uniformly bounded and collisionless solutions.

MSC:

70H07 Nonintegrable systems for problems in Hamiltonian and Lagrangian mechanics
70K55 Transition to stochasticity (chaotic behavior) for nonlinear problems in mechanics
70F05 Two-body problems
70G45 Differential geometric methods (tensors, connections, symplectic, Poisson, contact, Riemannian, nonholonomic, etc.) for problems in mechanics
70H40 Relativistic dynamics for problems in Hamiltonian and Lagrangian mechanics
37N05 Dynamical systems in classical and celestial mechanics
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