Relative equilibria in the 3-dimensional curved n-body problem

Abstract

We consider the 3-dimensional gravitational $n$-body problem, $n\ge 2$, in spaces of constant Gaussian curvature $\kappa \ne 0$, i.e.\ on spheres ${\mathbb{S}}_{\kappa }^3$, for $\kappa >0$, and on hyperbolic manifolds ${\mathbb{H}}_{\kappa }^3$, for $\kappa <0$. Our goal is to define and study relative equilibria, which are orbits whose mutual distances remain constant in time. We also briefly discuss the issue of singularities in order to avoid impossible configurations. We derive the equations of motion and define six classes of relative equilibria, which follow naturally from the geometric properties of ${\mathbb{S}}_{\kappa }^3$ and ${\mathbb{H}}_{\kappa }^3$. Then we prove several criteria, each expressing the conditions for the existence of a certain class of relative equilibria, some of which have a simple rotation, whereas others perform a double rotation, and we describe their qualitative behaviour. In particular, we show that in ${\mathbb{S}}_{\kappa }^3$ the bodies move either on circles or on Clifford tori, whereas in ${\mathbb{H}}_{\kappa }^3$ they move either on circles or on hyperbolic cylinders. Then we construct concrete examples for each class of relative equilibria previously described, thus proving that these classes are not empty. We put into the evidence some surprising orbits, such as those for which a group of bodies stays fixed on a great circle of a great sphere of ${\mathbb{S}}_{\kappa }^3$, while the other bodies rotate uniformly on a complementary great circle of another great sphere, as well as a large class of quasiperiodic relative equilibria, the first such non-periodic orbits ever found in a 3-dimensional $n$-body problem. Finally, we briefly discuss other research directions and the future perspectives in the light of the results we present here.

100 pages, 1 figure. arXiv admin note: text overlap with arXiv:0807.1747

References in corpus (4)

• An intrinsic approach in the curved n-body problem: the negative curvature case
• On the stability of tetrahedral relative equilibria in the positively curved 4-body problem
• Saari's Homographic Conjecture of the Three-Body Problem
• Levi-Civita regularization and geodesic flows for the `curved' Kepler problem

Cited by in corpus (20)

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• On the stability of tetrahedral relative equilibria in the positively curved 4-body problem
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• Bifurcations of the Lagrangian orbits from the classical to the curved 3-body problem
• Existence of a lower bound for the distance between point masses of relative equilibria for generalised quasi-homogeneous $n$-body problems and the curved $n$-body problem
• An integrable Henon-Heiles system on the sphere and the hyperbolic plane
• Rotopulsators of the curved N-body problem
• The Vlasov-Poisson System for Stellar Dynamics in Spaces of Constant Curvature
• Central configurations of the curved $N$-body problem
• The $N$-Body Problem in Spaces with Uniformly Varying Curvature
• Stability of fixed points and associated relative equilibria of the $3$-body problem on ${\mathbb{S}}^1$ and ${\mathbb{S}}^2$
• Möbius solutions of the curved $n$--body problem for positive curvature
• Three dimensional central configurations in H3 and S3
• Curvature as an integrable deformation
• Compactness and index of ordinary central configurations for the curved n-body problem
• Regular polygonal equilibrium configurations on S^1 and stability of the associated relative equilibria
• Polygonal negative hyperbolic rotopulsators of the curved $n$-body problem
• Equal masses results for choreographies $n$-body problems
• Positive elliptic-elliptic rotopulsators on Clifford tori of nonconstant size project onto regular polygons