Sergey V. Bolotin

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.

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.

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.

Minimization arguments are applied to a renormalized length functional to establish the existence of heteroclinic, homoclinic and chaotic geodesics on a two-dimensional torus satisfying some geometrical conditions.