A new method to study relative equilibria on \mathbbS²

2023/04/26 by Fujiwara, Toshiaki, Pérez-Chavela, Ernesto
#70F07 #70F10 #70F15 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.2304.13782

Abstract

We develop a new geometrical technique to study relative equilibria for a system of n--positive masses, moving on the two dimensional sphere \mathbbS², under the influence of a general potential which only depends on the mutual distances among the masses. The big difficulty to study relative equilibria on \mathbbS², that we call RE by short, is the absence of the center of mass as a first integral. We show that the two vanishing components of the angular momentum, for motions on \mathbbS², play the same role as the center of mass for motions on the Euclidean plane. From here we obtain that the rotation axis of a RE is one of the principal axes of the inertia tensor. Conditions for have RE and relations between the shape (given by the arc angles σ_ij among the masses) and the configuration (given by the polar angles θ_k and ϕ_i - ϕ_j in spherical coordinates) are shown. For n=3, we show explicitly the conditions to have Euler and Lagrange RE on \mathbbS². As an application of our method we study the the equal masses case for the positive curved three body problem where we show the existence of scalene and isosceles Euler RE and isosceles Lagrange RE.

Related

• Three-body relative equilibria on \mathbbS² I: Euler configurations
• Three-body relative equilibria on S² II: Extended Lagrangian configurations
• Three body relative equilibria on \mathbbS²
• Mass independent shapes for relative equilibria in the two dimensional constant positive curved three body problem
• Equal masses Eulerian relative equilibria on a rotating meridian of S²