Special solutions of the $n$-body problem: central configurations and choreographies

Abstract

[en] The $n$-body problem is a classical problem in celestial mechanics which attempts to describe the motion of $n$ bodies under their mutual gravitational attraction. The problem is only solvable for two masses, and not much is known for the general case of three or more bodies. This work deals with some particular solutions of the $n$-body problem. First, using its underlying Hamiltonian structure, we state the main properties of the problem, its symmetries and first integrals. Next, we study central configurations and their relation with homothetic and relative equilibria solutions. For three bodies, the well-known Lagrange configuration provides a relative equilibria in which three shifted particles in an equilateral triangle move along a periodic orbit, known as a choreography. In the last chapter we consider the figure eight solution which is another choreography of three bodies with some particular geometrical and dynamical properties. Using an ad-hoc implementation of the Taylor method developed for the numerical integration of the $n$-body problem we illustrate the orbits and the properties of the particular solutions discussed in this work as well as a numerical check of the remarkable linear stability property of the figure eight.