# memo_has_moved_text();Relative equilibria in the 3-dimensional curved $n$-body problem

Florin Diacu, Pacific Institute for the Mathematical Sciences. — and — Department of Mathematics and Statistics, University of Victoria, P.O. Box 3060 STN CSC, Victoria, BC, Canada, V8W 3R4

Publication: Memoirs of the American Mathematical Society
Publication Year: 2014; Volume 228, Number 1071
ISBNs: 978-0-8218-9136-0 (print); 978-1-4704-1483-2 (online)
DOI: https://doi.org/10.1090/memo/1071
Published electronically: July 22, 2013
Keywords: Celestial mechanics, $n$-body problems, spaces of constant curvature, fixed points, periodic orbits, quasiperiodic orbits, relative equilibria, qualitative behaviour of solutions.
MSC: Primary 70F10; Secondary 34C25, 37J45

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Chapters

• 1. Introduction
• 2. BACKGROUND AND EQUATIONS OF MOTION
• 3. ISOMETRIES AND RELATIVE EQUILIBRIA
• 4. CRITERIA AND QUALITATIVE BEHAVIOUR
• 5. EXAMPLES
• 6. CONCLUSIONS

### 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.

References