Hyperbolic motions of any shape exist in the Newtonian N-body problem

You need to know: Euclidean space $E={\mathbb R}^d$ of dimension $d$ , Euclidean norm $||a||$ of vector $a$ , (second) derivative of a function $x:{\mathbb R}\to E$ , small o notation $o(.)$ .

Background: Let N point particles with masses $m_i > 0$ and positions $x_i(t) \in E$ at time $t$ are moving according to Newton’s laws of motion: $m_j \frac{d^2 x_j}{dt^2} = \sum\limits_{i \neq j} \frac{m_i m_j(x_i - x j)}{r_{ij}^3}, \, 1 \leq j \leq N$ , where $r_{ij}$ is the distance between $x_i$ and $x_j$ . The motion of particles is determined by the initial conditions: their masses and their positions and velocities at time $t=0$ . Let $\Omega$ be the set of configurations such that the motion has no collisions ( $r_{ij}(t)>0$ for all $i,j$ and $t$ ). A motion is called hyperbolic if each particle has a different limit velocity vector, that is, $\lim\limits_{t\to \infty} \frac{d x_j}{dt}=a_j \in E$ and $a_i \neq a_j$ whenever $i \neq j$ .

The Theorem: On 25th August 2019, Ezequiel Maderna and Andrea Venturelli submitted to arxiv a paper in which they proved that for the Newtonian N-body problem in a space E of dimension $d\geq 2$ , there are hyperbolic motions $x:[0;+\infty) \to E^N$ such that $x(t) = \sqrt{2h} t a + o(t)$ as $t \to \infty$ for any choice of $x_0 = x(0) \in E^N$ , for any $a=(a_1, \dots, a_N) \in \Omega$ normalized by $||a||=1$ , and for any constant $h>0$ .

Short context: The problem of describing motion of N bodies under gravitation (N-body problem) in Euclidean space is a fundamental problem in physics and mathematics, studied by many authors, see, for example, here and here. In general, the motion can be very complicated even for $N=3$ , but can we at least understand hyperbolic motions? The only explicitly known hyperbolic motions are such that the shape of the configuration does not change with time, but it is conjectured that there are only finitely many such motions for any fixed $N$ . In contrast, the Theorem states that hyperbolic motions exist for all initial configurations and all choices of the limited velocities.