Sub-Finsler geometry and nonholonomic mechanics

TL;DR

This paper explores the geometric and mechanical properties of nonholonomic sub-Finsler structures, establishing conditions for extremals, invariance, and curvature, and comparing nonholonomic and vakonomic dynamics in a coordinate-free framework.

Contribution

It introduces the notion of nonholonomic sub-Finslerian structures, characterizes extremals, and relates nonholonomic mechanics to sub-Finsler geometry with a coordinate-free approach.

Findings

Conditions for abnormal and normal extremals are established.

Nonholonomic distributions are geodesically invariant under the Barthel connection.

The sub-Laplacian measures the curvature of the structure.

Abstract

We discuss a variational approach to the length functional and its relation to sub-Hamiltonian equations on sub-Finsler manifolds. Then, we introduce the notion of the nonholonomic sub-Finslerian structure and prove that the distributions are geodesically invariant concerning the Barthel non-linear connection. We provide necessary and sufficient conditions for the existence of the curves that are abnormal extremals; likewise, we provide necessary and sufficient conditions for normal extremals to be the motion of a free nonholonomic mechanical system, and vice versa. Moreover, we show that a coordinate-free approach for a free particle is a comparison between the solutions of the nonholonomic mechanical problem and the solutions of the Vakonomic dynamical problem for the nonholonomic sub-Finslerian structure. In addition, we provide an example of the nonholonomic sub-Finslerian structure. Finally, we show that the sub-Laplacian measures the curvature of the nonholonomic sub-Finslerian structure.