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Mat. Sb., 2010, Volume 201, Number 7, Pages 99–120 (Mi msb7617)  

This article is cited in 9 scientific papers (total in 9 papers)

Maxwell strata and symmetries in the problem of optimal rolling of a sphere over a plane

Yu. L. Sachkov

Program Systems Institute of RAS

Abstract: The problem of rolling of a sphere over a plane without slipping or twisting is considered. It is required to roll the sphere from one contact configuration into another one so that the curve traced by the contact point be of minimum length. Extremal trajectories in this problem were described by Arthur, Walsh and Jurdjevic.
In this work, discrete and continuous symmetries of the problem are constructed and fixed points of the action of these symmetries in the inverse image and image of the exponential map are studied. This analysis is used to derive necessary conditions for optimality; namely, upper bounds on the cut time along the extremal trajectories.
Bibliography: 21 titles.

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


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English version:
Sbornik: Mathematics, 2010, 201:7, 1029–1051

Bibliographic databases:

UDC: 517.977.52
MSC: Primary 70E15; Secondary 53C17, 93B29
Received: 04.08.2009 and 03.03.2010

Citation: Yu. L. Sachkov, “Maxwell strata and symmetries in the problem of optimal rolling of a sphere over a plane”, Mat. Sb., 201:7 (2010), 99–120; Sb. Math., 201:7 (2010), 1029–1051

Citation in format AMSBIB
\by Yu.~L.~Sachkov
\paper Maxwell strata and symmetries in the problem of optimal rolling of a~sphere over a~plane
\jour Mat. Sb.
\yr 2010
\vol 201
\issue 7
\pages 99--120
\jour Sb. Math.
\yr 2010
\vol 201
\issue 7
\pages 1029--1051

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    This publication is cited in the following articles:
    1. A. P. Mashtakov, Yu. L. Sachkov, “Extremal trajectories and the asymptotics of the Maxwell time in the problem of the optimal rolling of a sphere on a plane”, Sb. Math., 202:9 (2011), 1347–1371  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    2. A. A. Ardentov, Yu. L. Sachkov, “Extremal trajectories in a nilpotent sub-Riemannian problem on the Engel group”, Sb. Math., 202:11 (2011), 1593–1615  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    3. A. P. Mashtakov, “Asymptotics of extremal curves in the ball rolling problem on the plane”, Journal of Mathematical Sciences, 199:6 (2014), 687–694  mathnet  crossref  mathscinet
    4. Proc. Steklov Inst. Math., 278 (2012), 218–232  mathnet  crossref  mathscinet  isi  elib  elib
    5. I. Yu. Beschatnyi, “The optimal rolling of a sphere, with twisting but without slipping”, Sb. Math., 205:2 (2014), 157–191  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    6. A. Juhas, L. A. Novak, “Conflict set and waveform modelling for power amplifier design”, Math. Probl. Eng., 2015 (2015), 585962, 29 pp.  crossref  mathscinet  isi  elib  scopus
    7. A. A. Agrachev, “Topics in sub-Riemannian geometry”, Russian Math. Surveys, 71:6 (2016), 989–1019  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    8. Lazureanu C., Binzar T., “Symmetries and Properties of the Energy-Casimir Mapping in the Ball-Plate Problem”, Adv. Math. Phys., 2017, 5164602  crossref  mathscinet  isi  scopus
    9. A. A. Ardentov, Yu. L. Sachkov, T. Huang, X. Yang, “Extremal trajectories in the sub-Lorentzian problem on the Engel group”, Sb. Math., 209:11 (2018), 1547–1574  mathnet  crossref  crossref  adsnasa  isi  elib
  • Математический сборник Sbornik: Mathematics (from 1967)
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