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| Zero energy entire solutions to the full $N$-body problem with diverging radii are called parabolic solutions. In spite of of their natural structural instability, these orbits act as connections between different central configurations and can be used as carriers from one to the other region of the phase space. In the recent papers, we have linked the presence of minimal parabolic orbits with the existence of minimal collision trajectories and the detection of unbounded families of noncollision periodic orbits. The topologically non-trivial parabolic orbits are of interest also from the point of view of weak KAM theory, as they are homoclinic to the infinity, which represents the Aubry-Mather set of our system. They can be used to construct multiple viscosity solutions of the associated Hamilton-Jacobi equation.
We shall is to deal with the existence of homoclinic and heteroclinic trajectories linking central configurations of the full $N$-body problem, starting with the case when the configuration space is reduced by symmetries (platonic, dihedral). |
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