Super Central Configurations
by
Zhifu Xie
Department of Mathematics and Computer Science, Virginia State University, Petersburg, VA 23806

In this talk, we consider the inverse problem of central configurations of n-body problem. For a given q=(q₁, q₂, ..., q_n) ∈ (R^d)ⁿ, let S(q) be the admissible set of masses by

S(q)={ m=(m1, m2, ..., mn)| mi ∈ R+, q is a central configuration for m }.

We call m=(m₁, m₂, ..., m_n) ∈ S(q) and m'=(m₁', m₂', ..., m_n') ∈ S(q) equivalent if m=am' for some a ∈ R⁺ and let [S\tilde](q) be the set of equivalent classes in S(q). Denote by ^# [S\tilde](q) the cardinal number of equivalent elements for any given configuration q=(q₁, q₂, ..., q_n) ∈ (R^d)ⁿ. We call q a super central configuration if ^# [S\tilde](q) ≥ 2. The possible values of ^# [S\tilde](q) and the structure of the set S(q) are investigated for n ≤ 4. The existence of super central configuration for n > 4 is constructed. We prove that all central configurations are super central configurations for n ≤ 3 and no convex four-body central configuration is a super central configuration. We also prove that a super central configuration has an unusual property: a super central configuration gives rise to a perverse solution when center of mass of the configuration is fixed.

PDF

Date received: August 28, 2008