The angular momentum of a relative equilibrium

TL;DR

This paper investigates the angular momentum of relative equilibria in higher-dimensional Newtonian N-body systems, focusing on frequencies, bifurcations, and the role of complex structures, with full results in four dimensions and conjectures for higher.

Contribution

It provides a complete analysis of angular momentum frequencies and bifurcations for relative equilibria in four dimensions, extending understanding of higher-dimensional N-body dynamics.

Findings

Full characterization of angular momentum frequencies in 4D

Identification of bifurcation points from periodic to quasi-periodic motion

Conjecture and partial proof for higher-dimensional cases

Abstract

There are two main reasons why relative equilibria of N point masses under the influence of Newton attraction are mathematically more interesting to study when space dimension is at least 4: On the one hand, in a higher dimensional space, a relative equilibrium is determined not only by the initial configuration but also by the choice of a complex structure on the space where the motion takes place; in particular, its angular momentum depends on this choice; On the other hand, relative equilibria are not necessarily periodic: if the configuration is "balanced" but not central, the motion is in general quasi-periodic. In this exploratory paper we address the following question, which touches both aspects: what are the possible frequencies of the angular momentum of a given central (or balanced) configuration and at what values of these frequencies bifurcations from periodic to quasi-periodic relative equilibria do occur ? We give a full answer for relative equilibrium motions in dimension 4 and conjecture that an analogous situation holds true for higher dimensions. A refinement of Horn's problem given by Fomin, Fulton, Li and Poon plays an important role. P.S. The conjecture is now proved (see Alain Chenciner and Hugo Jimenez Perez, Angular momentum and Horn's problem, arXiv:1110.5030v1 [math.DS]).