Continuous Families of Central Configurations for Even-Power Homogeneous Potentials
Abstract
We study central configurations for n bodies interacting through pair potentials Uσ = κ/σ ∑i<jmi mj rijσ, 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, k≥2, 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.
Related
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- Finiteness of planar central configurations of five bodies with positive masses
- Sharp Weighted Hardy-Type Inequalities for Homogeneous Magnetic Potentials
- Planar Central Configurations of Four Equal Masses: Classification and Morse Indices
- Uniqueness of four-body convex central configurations for all masses
BibTeX
@misc{nurowski2026continuous,
title = {{Continuous Families of Central Configurations for Even-Power Homogeneous Potentials}},
author = {Pawel Nurowski},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.04136},
eprint = {2610.04136},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.04136},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Continuous Families of Central Configurations for Even-Power Homogeneous Potentials
AU - Pawel Nurowski
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.04136
UR - https://arxiv.org/abs/2610.04136
AB - We study central configurations for n bodies interacting through pair potentials U_σ = κ/σ ∑_i<jm_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, k≥2, 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.
ER -