Shadowing chains of collision orbits
Sergey V. Bolotin
Discrete & Continuous Dynamical Systems - A 2006, 14(2): 235-260 doi: 10.3934/dcds.2006.14.235
We consider a Hamiltonian system modeling the plane restricted elliptic 3 body problem with one of the masses small and prove the existence of periodic and chaotic orbits shadowing chains of collision orbits. Periodic orbits of this type were first studied by Poincaré for the non-restricted 3 body problem. The present paper contains general results which hold for time periodic Hamiltonian systems with a small Newtonian singularity. Applications to celestial mechanics will be given in a subsequent paper.
keywords: action functional shadowing orbit. Three-body problem
Global regularization for the $n$-center problem on a manifold
Sergey V. Bolotin Piero Negrini
Discrete & Continuous Dynamical Systems - A 2002, 8(4): 873-892 doi: 10.3934/dcds.2002.8.873
We describe a global version of the KS regularization of the $n$-center problem on a closed 3-dimensional manifold. The regularized configuration manifold turns out to be 4 or 5 dimensional closed manifold depending on whether $n$ is even or odd. As an application, we show that the $n$ center problem in $S^3$ has positive topological entropy for $n\ge 5$ and energy greater than the maximum of the potential energy. The proof is based on the results of Gromov and Paternain on the topological entropy of geodesic flows. This paper is a continuation of [6], where global regularization of the $n$-center problem in $\mathbf R^3$ was studied.
keywords: topological entropy. Newtonian singularity Lagrangian system regularization
Chaotic trajectories for natural systems on a torus
M. L. Bertotti Sergey V. Bolotin
Discrete & Continuous Dynamical Systems - A 2003, 9(5): 1343-1357 doi: 10.3934/dcds.2003.9.1343
We consider a natural Lagrangian system on a torus and give sufficient conditions for the existence of chaotic trajectories for energy values slightly below the maximum of the potential energy. It turns out that chaotic trajectories always exist except when the system is "variationally separable", i.e. minimizers of the action functional behave like in a separable system. This gives some more support for an old conjecture that only separable natural Lagrangian systems on a torus are integrable.
keywords: Natural Lagrangian systems shadowing trajectory. Maupertuis principle homoclinic orbit chaotic trajectory Jacobi metric
Variational approach to second species periodic solutions of Poincaré of the 3 body problem
Sergey V. Bolotin Piero Negrini
Discrete & Continuous Dynamical Systems - A 2013, 33(3): 1009-1032 doi: 10.3934/dcds.2013.33.1009
We consider the plane 3 body problem with 2 of the masses small. Periodic solutions with near collisions of small bodies were named by Poincaré second species periodic solutions. Such solutions shadow chains of collision orbits of 2 uncoupled Kepler problems. Poincaré only sketched the proof of the existence of second species solutions. Rigorous proofs appeared much later and only for the restricted 3 body problem. We develop a variational approach to the existence of second species periodic solutions for the nonrestricted 3 body problem. As an application, we give a rigorous proof of the existence of a class of second species solutions.
keywords: collision. action functional 3 body problem periodic orbit

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