Platonic constellations of periodic motions in the (n + 1)-body problem
Abstract
We study the spatial (n+1)-body problem formed by one heavy central mass together with n equal masses placed on a single orbit of a polyhedral rotation group H∈{T,O,I}, so that n=|H|∈{12,24,60}. Imposing the symmetry qL=L qI for L∈ H reduces the problem to a single 2π-periodic reference curve, with reduced action AH=A0+ε A1, in which ε is the inverse central mass and A0 is the Kepler action. At ε=0 the critical set contains, as one connected component, the five-dimensional manifold of Kepler ellipses of minimal period 2π, which we prove to be a nondegenerate critical manifold. A Lyapunov--Schmidt reduction along this manifold turns the continuation problem into the search for nondegenerate critical points of an explicit function Φ(e,ψ) of the eccentricity e and the spatial orientation ψ, a nondegeneracy we verify by a computer-assisted proof. We thereby obtain, for each of the three groups, families of periodic solutions of the (n+1)-body problem with n+1∈{13,25,61} bodies, bifurcating from Kepler ellipses and carrying the full tetrahedral, octahedral, or icosahedral symmetry.
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BibTeX
@misc{constantineau2026platonic,
title = {{Platonic constellations of periodic motions in the $(n + 1)$-body problem}},
author = {Kevin Constantineau and Carlos García-Azpeitia and Jean-Philippe Lessard},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10242},
eprint = {2609.10242},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10242},
}
RIS
TY - UNPB
TI - Platonic constellations of periodic motions in the (n + 1)-body problem
AU - Kevin Constantineau
AU - Carlos García-Azpeitia
AU - Jean-Philippe Lessard
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.10242
UR - https://arxiv.org/abs/2609.10242
AB - We study the spatial (n+1)-body problem formed by one heavy central mass together with n equal masses placed on a single orbit of a polyhedral rotation group H∈{T,O,I}, so that n=|H|∈{12,24,60}. Imposing the symmetry q_L=L q_I for L∈ H reduces the problem to a single 2π-periodic reference curve, with reduced action A_H=A_0+ε A_1, in which ε is the inverse central mass and A_0 is the Kepler action. At ε=0 the critical set contains, as one connected component, the five-dimensional manifold of Kepler ellipses of minimal period 2π, which we prove to be a nondegenerate critical manifold. A Lyapunov--Schmidt reduction along this manifold turns the continuation problem into the search for nondegenerate critical points of an explicit function Φ(e,ψ) of the eccentricity e and the spatial orientation ψ, a nondegeneracy we verify by a computer-assisted proof. We thereby obtain, for each of the three groups, families of periodic solutions of the (n+1)-body problem with n+1∈{13,25,61} bodies, bifurcating from Kepler ellipses and carrying the full tetrahedral, octahedral, or icosahedral symmetry.
ER -