Diacu, Florin; Santoprete, Manuele Nonintegrability and chaos in the anisotropic Manev problem. (English) Zbl 0996.70015 Physica D 156, No. 1-2, 39-52 (2001). 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. Cited in 12 Documents 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 Keywords:nonintegrability; extended Poincaré-Melnikov method; anisotropic Manev problem; Newtonian force law; relativistic correction; weak anisotropy; chaos; zero-energy manifold; isolated periodic orbits; geodesic deviation approach PDFBibTeX XMLCite \textit{F. Diacu} and \textit{M. Santoprete}, Physica D 156, No. 1--2, 39--52 (2001; Zbl 0996.70015) Full Text: DOI arXiv