Parabolic solutions for the planar $N$-centre problem: multiplicity and scattering

Abstract

For the planar $N$-centre problem $$\ddot{x}=-\sum _{i=1}^N\frac{m_i(x-c_i)}{|x-c_i{|}^{\alpha +2}},x\in {\mathbb{R}}^2\setminus \{c_1,\dots ,c_N\},$$ where $m_i>0$ for $i=1,\dots ,N$ and $\alpha \in [1,2)$, we prove the existence of entire parabolic trajectories, having prescribed asymptotic directions for $t\to \pm \infty$ and prescribed topological characterization with respect to the set of the centres.

References in corpus (2)

• Globally minimizing parabolic motions in the Newtonian N-body problem
• Scattering parabolic solutions for the spatial N-centre problem

Cited by in corpus (4)

• Periodic solutions to a forced Kepler problem in the plane
• 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