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Equilibria in non-Euclidean geometries

Z. Lángi, S. Wang

TL;DR

This work generalizes centroids and equilibrium theory to spaces of constant curvature and to normed geometries, connecting to Euclidean results on equilibrium counts. It defines centroids via first moments in spherical, hyperbolic, and normed spaces and classifies equilibria using appropriate distance notions, extended by Poincaré–Hopf constraints. The authors prove that every plane convex body in a plane of constant curvature or in a smooth 2D normed plane has at least four equilibria, and construct mono-monostatic 3D bodies in $ S^3$, $ H^3$, and certain rotationally symmetric normed spaces using projection-based deformations and moment balancing. The innovative use of gnomonic/projective-ball projections, two-parameter deformation families, and first-moment balancing provides a path toward mono-monostatic configurations in broader non-Euclidean geometries and normed settings.

Abstract

In this paper, extending the work of Gal'perin (Comm. Math. Phys. 154: 63-84, 1993), we investigate generalizations of the concepts of centroids and static equilibrium points of a convex body in spherical, hyperbolic and normed spaces. In addition, we examine the minimum number of equilibrium points a $2$- or $3$-dimensional convex body can have in these spaces. In particular, we show that every plane convex body in any of these spaces has at least four equilibrium points, and that there are mono-monostatic convex bodies in $3$-dimensional spherical, hyperbolic, and certain normed spaces. Our results are generalizations of results of Domokos, Papadopoulos and Ruina (J. Elasticity 36: 59-66, 1994), and Várkonyi and Domokos (J. Nonlinear Sci. 16: 255-281, 2006) for convex bodies in Euclidean space.

Equilibria in non-Euclidean geometries

TL;DR

This work generalizes centroids and equilibrium theory to spaces of constant curvature and to normed geometries, connecting to Euclidean results on equilibrium counts. It defines centroids via first moments in spherical, hyperbolic, and normed spaces and classifies equilibria using appropriate distance notions, extended by Poincaré–Hopf constraints. The authors prove that every plane convex body in a plane of constant curvature or in a smooth 2D normed plane has at least four equilibria, and construct mono-monostatic 3D bodies in , , and certain rotationally symmetric normed spaces using projection-based deformations and moment balancing. The innovative use of gnomonic/projective-ball projections, two-parameter deformation families, and first-moment balancing provides a path toward mono-monostatic configurations in broader non-Euclidean geometries and normed settings.

Abstract

In this paper, extending the work of Gal'perin (Comm. Math. Phys. 154: 63-84, 1993), we investigate generalizations of the concepts of centroids and static equilibrium points of a convex body in spherical, hyperbolic and normed spaces. In addition, we examine the minimum number of equilibrium points a - or -dimensional convex body can have in these spaces. In particular, we show that every plane convex body in any of these spaces has at least four equilibrium points, and that there are mono-monostatic convex bodies in -dimensional spherical, hyperbolic, and certain normed spaces. Our results are generalizations of results of Domokos, Papadopoulos and Ruina (J. Elasticity 36: 59-66, 1994), and Várkonyi and Domokos (J. Nonlinear Sci. 16: 255-281, 2006) for convex bodies in Euclidean space.
Paper Structure (9 sections, 15 theorems, 41 equations, 4 figures)

This paper contains 9 sections, 15 theorems, 41 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Centroids in non-Euclidean spaces
  4. Equilibria in non-Euclidean spaces
  5. The proof of Theorem \ref{['thm:2d']}
  6. The proof of Theorem \ref{['thm:3d']}
  7. The case $\mathbb{V}^3 = \mathbb{S}^3$
  8. The case $\mathbb{V}^3 = \mathbb{H}^3$
  9. The case when $\mathbb{V}^3$ is a rotationally symmetric $3$-dimensional normed space

Key Result

Theorem 1.1

Let $\mathbb{V}^2$ be a plane of constant curvature, or a normed plane with a smooth and strictly convex unit disk. Let $K$ be a plane convex body in $\mathbb{V}$. Then $K$ has at least four equilibrium points.

Figures (4)

  • Figure 1: Illustration of the function $F_c(x)$ for different values of $c$. Notation: continuous line for $c=1$, long dashed line for $c=0.3$, dashed line for $c=0.1$, dashed-dotted line for $c=0.03$, and dotted line for $c=0.01$.
  • Figure 2: Diagram of $a_c(\theta,\varphi)$ versus $\varphi$ for two different values of $c$. If $c=1$ then $a_c=\cos(2\varphi)$ for all values of $\theta$ (left panel), whereas for other values of $c$ and $\theta\neq 0$, it becomes somewhat distorted. The right panel shows the values of $a_c(\theta,\varphi)$ for $c=0.1$, with $\theta=-0.45\pi$ (continuous line), $\theta=-0.225 \theta$ (long dashed line), $\theta=0$ (dashed line), $\theta=0.225\pi$ (dashed-dotted line), and $\theta=0.45\pi$ (dotted line).
  • Figure 3: Contour plot of $\rho_{0.1}(\theta,\varphi)$. Darker color denotes smaller value of the function.
  • Figure 4: The set $K(0.1,d)$ for some values of $d$. Left panel: $d=0.4$, middle panel: $d=0.1$, right panel: $d=0.05$.

Theorems & Definitions (30)

  • Theorem 1.1
  • Theorem 1.2
  • Conjecture 1
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Remark 2.6
  • Remark 2.7
  • ...and 20 more