Finsler and sub-Finsler geodesics with chattering

2025/05/30 by L. V. Lokutsievskiy, Lokutsievskiy, L. V., M. I. Zelikin +1
Physics and Astronomy · #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Metric Geometry (math.MG) #Optimization and Control (math.OC)

paper · pdf · doi:10.48550/arxiv.2505.24474

openalex publication_date 2025/05/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Normal Curves in Sub-Finsler Lie Groups: Branching for Strongly Convex Norms and Face Stability for Polyhedral Norms
• Metric Lie Groups. Carnot-Carathéodory spaces from the homogeneous viewpoint
• Extremals for a series of sub-Finsler problems with 2-dimensional control via convex trigonometry
• Sub-Finsler geodesics on the Cartan group
• Sub-Finsler structures from the time-optimal control viewpoint for some nilpotent distributions
• Typicality of Chaotic Fractal Behavior of Integral Vortices in Hamiltonian Systems with Discontinuous Right Hand Side
• Croissance des boules et des géodésiques fermées dans les nilvariétés

Related

• Sub-Finsler geodesics on the Cartan group
• Blowups and blowdowns of geodesics in Carnot groups
• Infinite geodesics of sub-Finsler distances in the Heisenberg groups
• The optimal lower bound estimation of the number of closed geodesics on Finsler compact space form S²ⁿ⁺¹/ Γ
• A sub-Finsler problem on the Cartan group