Continuous Families of Central Configurations for Even-Power Homogeneous Potentials

Abstract

We study central configurations for $n$ bodies interacting through pair potentials \[ U_σ = \fracκσ \sum_{i<j}m_i m_j r_{ij}^σ, \] with logarithmic limit at $σ=0$. For the Newtonian exponent $σ=-1$, the classical Chazy--Wintner--Smale finiteness problem asks whether, for prescribed positive masses, there are only finitely many central configurations up to similarity. Finiteness is known in several important special cases and under additional dimensional or genericity hypotheses. It is therefore natural to ask whether an analogous finiteness principle continues to hold for other homogeneous exponents. We show that for every positive even exponent \[ σ=2k, \qquad k\ge2, \] the answer is emphatically negative. In fact, these exponents admit an abundance of continuous families of pairwise non-similar central configurations. The mechanism is provided by spherical designs: on a common sphere the force kernel for $σ=2k$ restricts to a polynomial of degree at most $k$, and consequently every positive weighted spherical $k$-design is a central configuration. We compute the corresponding central multiplier explicitly.