Rotopulsators of the curved N-body problem

TL;DR

This paper investigates special rotating and size-changing solutions called rotopulsators in the curved N-body problem, classifying their types, establishing existence criteria, and analyzing their qualitative behavior in spaces of positive and negative curvature.

Contribution

It introduces a comprehensive classification of rotopulsators, derives existence criteria and conservation laws, and proves new results on their qualitative behavior in curved spaces.

Findings

Five types of rotopulsators identified based on curvature and rotation.

Existence criteria established for each rotopulsator type.

No foliation of the 3-sphere with Clifford tori confines bodies in positive curvature.

Abstract

We consider the N-body problem in spaces of constant curvature and study its rotopulsators, i.e.\ solutions for which the configuration of the bodies rotates and changes size during the motion. Rotopulsators fall naturally into five groups: positive elliptic, positive elliptic-elliptic, negative elliptic, negative hyperbolic, and negative elliptic-hyperbolic, depending on the nature and number of their rotations and on whether they occur in spaces of positive or negative curvature. After obtaining existence criteria for each type of rotopulsator, we derive their conservation laws. We further deal with the existence and uniqueness of some classes of rotopulsators in the 2- and 3-body case and prove two general results about the qualitative behaviour of rotopulsators. More precisely, for positive curvature we show that there is no foliation of the 3-sphere with Clifford tori such that the motion of each body is confined to some Clifford torus. For negative curvature, a similar result is proved relative to foliations of the hyperbolic 3-sphere with hyperbolic cylinders.