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Mat. Sb., 2014, Volume 205, Number 2, Pages 3–38 (Mi msb8296)  

This article is cited in 1 scientific paper (total in 1 paper)

The optimal rolling of a sphere, with twisting but without slipping

I. Yu. Beschatnyi

Program Systems Institute of RAS, Yaroslavskaya obl., Pereslavskii raion, s. Ves'kovo

Abstract: The problem of a sphere rolling on the plane, with twisting but without slipping, is considered. It is required to roll the sphere from one configuration to another in such a way that the minimum of the action is attained. We obtain a complete parametrization of the extremal trajectories and analyse the natural symmetries of the Hamiltonian system of the Pontryagin maximum principle (rotations and reflections) and their fixed points. Based on the estimates obtained for the fixed points we prove upper estimates for the cut time, that is, the moment of time when an extremal trajectory loses optimality. We consider the problem of re-orienting the sphere in more detail; in particular, we find diffeomorphic domains in the pre-image and image of the exponential map which are used to construct the optimal synthesis.
Bibliography: 15 titles.

Keywords: optimal control, geometric methods, symmetries, rolling of surfaces.

Funding Agency Grant Number
Ministry of Education and Science of the Russian Federation 14.B25.31.0029
Russian Foundation for Basic Research 12-01-00913
13-01-91162-ГФЕН_а


DOI: https://doi.org/10.4213/sm8296

Full text: PDF file (848 kB)
References: PDF file   HTML file

English version:
Sbornik: Mathematics, 2014, 205:2, 157–191

Bibliographic databases:

UDC: 517.538
PACS: 45.80.+r
MSC: Primary 49K15; Secondary 70B10, 93B27
Received: 28.10.2013

Citation: I. Yu. Beschatnyi, “The optimal rolling of a sphere, with twisting but without slipping”, Mat. Sb., 205:2 (2014), 3–38; Sb. Math., 205:2 (2014), 157–191

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. A. Agrachev, “Topics in sub-Riemannian geometry”, Russian Math. Surveys, 71:6 (2016), 989–1019  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
  • Математический сборник Sbornik: Mathematics (from 1967)
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