Hooke and Coulomn Energy of Tripod Spiders

TL;DR

This paper analyzes the energy landscapes of tripod spiders under Hooke and Coulomb potentials, providing a complete Morse theory description and exploring robust control domains related to classical physics conjectures.

Contribution

It offers a comprehensive Morse theory analysis for Hooke potential and studies the robust control domain for Coulomb energy in tripod spiders, connecting to the Maxwell conjecture.

Findings

Complete Morse theory description for Hooke potential

Non-empty robust control domain for Coulomb energy with positive charges

Connections to Maxwell's conjecture about point charges

Abstract

Tripod spiders are the simplest examples of arachnoid mechanisms. Their workspaces and configuration spaces are well known. For Hooke potential, we give a complete description of the Morse theory and treat the robust control of the spider. For the Coulomb energy, we use stationary charges and the trapping domain to study the robust control of spiders. We show that, for a regular triangle and positive charges, the domain of robust control is non-void. This relates to questions about the Maxwell conjecture about point charges. We end with several natural problems and research perspectives suggested by our results.