Finsler and sub-Finsler geodesics with chattering

TL;DR

This paper demonstrates the existence of geodesics with chattering behavior in Finsler and sub-Finsler manifolds, providing explicit examples and conditions that lead to such complex geodesic switching phenomena.

Contribution

It offers the first explicit examples of chattering geodesics in Finsler and sub-Finsler geometries and answers a key open question negatively.

Findings

Existence of chattering geodesics in Finsler and sub-Finsler manifolds

Explicit construction on a Carnot group showing chattering behavior

A sufficient condition for chattering in Pontryagin extremals

Abstract

In this paper, we provide examples of Finsler and sub-Finsler manifolds whose geodesics exhibit chattering, that is, a countable number of switches over an arbitrarily small time interval. We also present an explicit left-invariant structure on a Carnot group whose geodesics exhibit chattering. This provides a negative answer to Le Donne's question. Furthermore, the paper presents a sufficient condition for normal Pontryagin maximum principle extremals in (sub-)Finsler problems to exhibit chattering.