Convexity Analysis of Snake Models Based on Hamiltonian Formulation

TL;DR

This paper analyzes the convexity properties of dynamic snake models using Hamiltonian mechanics, establishing conditions for attractor stability and linking energy convexity to model dynamics.

Contribution

It introduces a convexity analysis framework for snake models based on Hamiltonian formulation, connecting energy properties to dynamic stability.

Findings

Convexity of the potential energy is necessary for attractor stability.

A local relation between dynamic parameters and convergence rate is derived.

Hamiltonian analysis relates physical energy to the snake model's dynamics.

Abstract

This paper presents a convexity analysis for the dynamic snake model based on the Potential Energy functional and the Hamiltonian formulation of the classical mechanics. First we see the snake model as a dynamical system whose singular points are the borders we seek. Next we show that a necessary condition for a singular point to be an attractor is that the energy functional is strictly convex in a neighborhood of it, that means, if the singular point is a local minimum of the potential energy. As a consequence of this analysis, a local expression relating the dynamic parameters and the rate of convergence arises. Such results link the convexity analysis of the potential energy and the dynamic snake model and point forward to the necessity of a physical quantity whose convexity analysis is related to the dynamic and which incorporate the velocity space. Such a quantity is exactly the (conservative) Hamiltonian of the system.