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Math & Stats Department Colloquium

Friday, October 6th, 2023

Time: 2:30 p.m.  Place: Jeffery Hall, Room 234

Speaker: Ernesto Pérez-Chavela (ITAM)

Title: A walk with relative equilibria from the plane to the sphere $\mathbb{S}^2$

Abstract: The simplest solutions of the $N$–body problem are those where the mutual distances among the masses remain constant for all time, that is, the motions behave as a rigid body. For $N = 3$ on the Euclidean plane it is well known that there are exactly five relative equilibria: three collinear (Euler relative equilibria) and two planar forming an equilateral triangle (Lagrange relative equilibria).

In this talk, quickly I will describe the above relative equilibria, and I extend this concept to the sphere $\mathbb{S}^2$. The big difficulty to study relative equilibria on the sphere $\mathbb{S}^2$, that we call RE by short, is the absence of the center of mass as a first integral, since many of the standard methods used in the classical case don’t apply any more. Without the center of mass we do not know how to determine the rotation axis. I will show a geometrical method to study relative equilibria on the sphere (RE by short). We assume that the masses are moving under the influence of a general potential which only depends on the mutual distances among the masses. First we prove the existence of two new integrals of motion, which can be seen as an extension of the center of mass. These two new integrals allow us determine the rotation axis. For simplicity in the computations, we restrict our analysis to the case $N = 3$. Applying our method, we give some new families of Euler and Lagrange RE on the sphere for the cotangent potential (the natural extension of the Newtonian potential to the sphere).

Bio: Prof. Pérez-Chavela received his PhD from the Universidad Autónoma Metropolitana-Iztapalapa in 1991, and is now a professor and an emeritus national researcher at Instituto Tecnológico Autónomo de México (ITAM) in Mexico. His main research interests center around Hamiltonian systems, celestial mechanics, as well as more general dynamical systems and the qualitative theory of ODE.