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### Open Access Journals

DCDS

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.

DCDS

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.

DCDS

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.

DCDS

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.

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