Static Equilibria of Perturbed Spheres: A Single-Harmonic Class Map, a Parity Obstruction, and a Certified Counter for the Mono-Monostatic Regime
Abstract
Varkonyi and Domokos proved that homogeneous convex bodies with any prescribed numbers S≥1 of stable and U≥1 of unstable static equilibria exist, the case S=U=1 being the mono-monostatic Gomboc. We give a complete constructive answer for the simplest nondegenerate shapes: a homogeneous body whose boundary is the unit sphere perturbed radially by a single real spherical harmonic Yℓm of degree ℓ≥2 and order 1≤ m≤ℓ has exactly S=U=m(ℓ−m+1) stable and unstable equilibria, for every amplitude in the convex range when m≥2, and for small amplitude when m=1. The reduction to the critical points of the perturbing harmonic is an identity when m≥2: the symmetry of a single tesseral harmonic pins the centroid at the origin exactly, so the centroid-to-surface distance is a strictly increasing function of the harmonic. We count the critical points of Yℓm from its separable structure and verify the Poincare-Hopf balance in index form. Three consequences follow. (i) A single harmonic populates only the diagonal S=U of the equilibrium-class lattice; no single harmonic of degree ≥2 is mono-monostatic. (ii) Any centrally symmetric (even-degree) perturbation has even S and U and cannot be mono-monostatic, a parity obstruction. (iii) Mono-monostaticity is therefore intrinsically multi-harmonic. A predict-then-confirm numerical study on eight single-harmonic bodies matches the formula exactly. For the multi-harmonic regime we give a certified equilibrium counter (interval arithmetic with Krawczyk uniqueness and Hessian classification) that, given the centroid, provably neither under- nor over-counts. It certifies that a member of a known analytic family, at an amplitude of our own choosing inside the strictly convex sub-regime, is mono-monostatic. For that body the centroid is exactly the origin, so its certificate is unconditional.
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BibTeX
@misc{couey2026static,
title = {{Static Equilibria of Perturbed Spheres: A Single-Harmonic Class Map, a Parity Obstruction, and a Certified Counter for the Mono-Monostatic Regime}},
author = {Vincent Wesley Couey},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2608.11213},
eprint = {2608.11213},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.11213},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Static Equilibria of Perturbed Spheres: A Single-Harmonic Class Map, a Parity Obstruction, and a Certified Counter for the Mono-Monostatic Regime
AU - Vincent Wesley Couey
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2608.11213
UR - https://arxiv.org/abs/2608.11213
AB - Varkonyi and Domokos proved that homogeneous convex bodies with any prescribed numbers S≥1 of stable and U≥1 of unstable static equilibria exist, the case S=U=1 being the mono-monostatic Gomboc. We give a complete constructive answer for the simplest nondegenerate shapes: a homogeneous body whose boundary is the unit sphere perturbed radially by a single real spherical harmonic Y_ℓ^m of degree ℓ≥2 and order 1≤ m≤ℓ has exactly S=U=m(ℓ−m+1) stable and unstable equilibria, for every amplitude in the convex range when m≥2, and for small amplitude when m=1. The reduction to the critical points of the perturbing harmonic is an identity when m≥2: the symmetry of a single tesseral harmonic pins the centroid at the origin exactly, so the centroid-to-surface distance is a strictly increasing function of the harmonic. We count the critical points of Y_ℓ^m from its separable structure and verify the Poincare-Hopf balance in index form. Three consequences follow. (i) A single harmonic populates only the diagonal S=U of the equilibrium-class lattice; no single harmonic of degree ≥2 is mono-monostatic. (ii) Any centrally symmetric (even-degree) perturbation has even S and U and cannot be mono-monostatic, a parity obstruction. (iii) Mono-monostaticity is therefore intrinsically multi-harmonic. A predict-then-confirm numerical study on eight single-harmonic bodies matches the formula exactly. For the multi-harmonic regime we give a certified equilibrium counter (interval arithmetic with Krawczyk uniqueness and Hessian classification) that, given the centroid, provably neither under- nor over-counts. It certifies that a member of a known analytic family, at an amplitude of our own choosing inside the strictly convex sub-regime, is mono-monostatic. For that body the centroid is exactly the origin, so its certificate is unconditional.
ER -