Radial kinetic nonholonomic trajectories are Riemannian geodesics!

descriptionPublicationkeyboard_double_arrow_right Article , Preprint 13 Aug 2021Embargo end date: 01 Jan 2020 English Publisher:Springer Science and Business Media LLCJournal:Analysis and Mathematical Physics, volume 11 (issn: 1664-2368, eissn: 1664-235X,

Copyright policy )Funded by:FCT | Análise Geométrica e Numé...

Authors: Alexandre Anahory Simoes; Juan Carlos Marrero; David Martín de Diego;

doi: 10.1007/s13324-021-00584-3 , 10.48550/arxiv.2010.12444

arXiv: 2010.12444

handle: 10261/255347

Radial kinetic nonholonomic trajectories are Riemannian geodesics!

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Abstract

AbstractNonholonomic mechanics describes the motion of systems constrained by nonintegrable constraints. One of its most remarkable properties is that the derivation of the nonholonomic equations is not variational in nature. However, in this paper, we prove (Theorem 1.1) that for kinetic nonholonomic systems, the solutions starting from a fixed pointqare true geodesics for a family of Riemannian metrics on the image submanifold$${{\mathcal {M}}}^{nh}_q$$Mqnhof the nonholonomic exponential map. This implies a surprising result: the kinetic nonholonomic trajectories with starting pointq, for sufficiently small times, minimize length in$${{\mathcal {M}}}^{nh}_q$$Mqnh!

Related Organizations

University of La Laguna
Spain
Universidad de La Laguna (ULL)
Spain
Institute of Mathematical Sciences
Spain
Spanish National Research Council
Spain

Keywords

nonholonomic mechanics, Mathematics - Differential Geometry, Nonholonomic systems related to the dynamics of a system of particles, radial geodesics, FOS: Physical sciences, Methods of global Riemannian geometry, including PDE methods; curvature restrictions, Mathematical Physics (math-ph), 70G45 (Primary), 53B20, 53C21, 37J60, 70F25 (Secondary), nonholonomic geodesics, exponential map, Local Riemannian geometry, Differential geometric methods (tensors, connections, symplectic, Poisson, contact, Riemannian, nonholonomic, etc.) for problems in mechanics, Differential Geometry (math.DG), FOS: Mathematics, Riemannian geometry, Nonholonomic dynamical systems, Mathematical Physics

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