On the stability of periodic N-body motions with the symmetry of Platonic polyhedra

Abstract

In [13] several periodic orbits of the Newtonian N -body problem have been found as minimizers of the Lagrangian action in suitable sets of T -periodic loops, for a given T > 0. Each of them share the symmetry of one Platonic polyhedron. In this paper we first present an algorithm to enumerate all the orbits that can be found following the proof in [13]. Then we describe a procedure aimed to compute them and study their stability. Our computations suggest that all these periodic orbits are unstable. For some cases we produce a computerassisted proof of their instability using multiple precision interval arithmetic.

Braids with the symmetries of Platonic polyhedra in the Coulomb (N+1)-body problem

“…The main method is a variant of the well-known shooting method, but here problems arise when we search for a good starting guess. In [11] the starting guess was computed using a gradient descent method, applied to the discretized version of the action functional of the N -body problem. In our case we cannot use the same method, because the lack of coercivity of the action (5) leads to a failure of the gradient method, that we experienced in our numerical experiments.…”

“…In [11,13] periodic orbits of the N -body problem with equal masses, satisfying these symmetries and topological constraints, were found as minimizers of the Lagrangian action functional, using Calculus of Variations techniques in order to prove their existence. In our case, taking into account the symmetry (a), the action functional writes as…”

“…In more recent years, the discovery of the Figure Eight solution of the three-body problem [20] started a research on particular periodic orbits of the N -body problem, that now are known with the name of choreographies. Numerical evidence of the existence of such orbits are given in [25][26][27], and also rigorous computer assisted methods to study their existence and their dynamical properties have been used [11,[15][16][17]. The first rigorous proof of the existence of the Figure Eight orbit was given in [5], and it uses the method of minimization of the action over a particular set of periodic loops.…”

“…We search for periodic motions sharing the symmetry of Platonic polyhedra, that is Tetrahedron, Cube, Octahedron, Dodecahedron and Icosahedron, hence N can be either 12, 24 or 60. For the Newtonian N -body problem, a list of orbits with this symmetry was found in [11]. Here we have been able to compute a similar set of periodic orbits, see the webpage [9] for animations of these solutions.…”

“…Our numerical computations also show that these orbits are unstable. Moreover, since the approach used in [11,13] for the proof of the existence was the minimization of the action, here we investigated also if this method could still work for the Coulomb (N + 1)-body problem. However, we show numerically that the orbits we compute are not minimizers of the action, not even locally.…”