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arXiv · 2309.03489

Sub-Finsler geometry and nonholonomic mechanics

Layth M. Alabdulsada
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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.

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Layth M. Alabdulsada. 2023-09-07. Sub-Finsler geometry and nonholonomic mechanics. https://doi.org/10.1216/rmj.2025.55.597

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