Scattering parabolic solutions for the spatial N-centre problem

Abstract

For the $N$-centre problem in the three dimensional space, $$\ddot{x}=-\sum _{i=1}^N\frac{m_i(x-c_i)}{|x-c_i{|}^{\alpha +2}},x\in {\mathbb{R}}^3\setminus \{c_1,\dots ,c_N\},$$ where $N\ge 2$, $m_i>0$ and $\alpha \in [1,2)$, we prove the existence of entire parabolic trajectories having prescribed asymptotic directions. The proof relies on a variational argument of min-max type. Morse index estimates and regularization techniques are used in order to rule out the possible occurrence of collisions.

References in corpus (6)

• Globally minimizing parabolic motions in the Newtonian N-body problem
• Entire minimal parabolic trajectories: the planar anisotropic Kepler problem
• On weak KAM theory for N-body problems
• Syzygies in the two center problem
• On the free time minimizers of the Newtonian N-body problem
• Chaotic quasi-collision trajectories in the 3-centre problem

Cited by in corpus (7)

• Periodic solutions to a forced Kepler problem in the plane
• Whiskered parabolic tori in the planar $(n+1)$-body problem
• Parabolic solutions for the planar $N$-centre problem: multiplicity and scattering
• Index Theory for Zero Energy Solutions of the Planar Anisotropic Kepler Problem
• Parabolic orbits in Celestial Mechanics: a functional-analytic approach
• Symbolic dynamics for the anisotropic $N$-centre problem at negative energies
• The spatial $N$-centre problem: scattering at positive energies