Application of Lagrangian mechanics equations for finding of the minimum distance between smooth submanifolds in N-dimensional Euclidean space -- Part II

TL;DR

This paper presents a method using Lagrangian mechanics and Riemannian geometry to find the minimal distance between smooth, non-crossing submanifolds in N-dimensional Euclidean space, emphasizing convergence properties.

Contribution

It introduces a dynamical systems approach based on Lagrangian mechanics to compute minimal distances, incorporating Lyapunov functions for convergence analysis.

Findings

Method effectively finds minimal distances between submanifolds.

Potential energy functions influence convergence of the dynamical system.

Framework leverages Riemannian geometry for geometric insights.

Abstract

The method of finding the minimal distance between smooth non crossing submanifolds in N-dimensional Euclidean space are presented. It based on solution of the equations that describe the dynamics of the pair of material points. The dynamical system can be presented as a natural mechanical system determined by Riemannian geometry on the manifold and chosen potential energy. Such an approach makes it possible to find Lyapunov function of the considered system and to formulates the requirements on the form of potential energy that brings to the convergence of the method.