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.
Citation:
Yu. L. Sachkov, “Maxwell strata and symmetries in the problem of optimal rolling of a sphere over a plane”, Sb. Math., 201:7 (2010), 1029–1051
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\paper Maxwell strata and symmetries in the problem of optimal rolling of a~sphere over a~plane
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\yr 2010
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\pages 1029--1051
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Linking options:
https://www.mathnet.ru/eng/sm7617
https://doi.org/10.1070/SM2010v201n07ABEH004101
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This publication is cited in the following 16 articles:
Yu. L. Sachkov, “Left-invariant optimal control problems on Lie groups that are integrable by elliptic functions”, Russian Math. Surveys, 78:1 (2023), 65–163
Seyed Amir Tafrishi, Mikhail Svinin, Motoji Yamamoto, Yasuhisa Hirata, “A geometric motion planning for a spin-rolling sphere on a plane”, Applied Mathematical Modelling, 121 (2023), 542
Yu. L. Sachkov, “Left-invariant optimal control problems on Lie groups: classification and problems integrable by elementary functions”, Russian Math. Surveys, 77:1 (2022), 99–163
Alexey Mashtakov, 2021 International Conference “Nonlinearity, Information and Robotics” (NIR), 2021, 1
A. V. Podobryaev, “Symmetries in left-invariant optimal control problems”, Sb. Math., 211:2 (2020), 275–290
A. V. Podobryaev, “Symmetric Extremal Trajectories in Left-Invariant Optimal Control Problems”, Rus. J. Nonlin. Dyn., 15:4 (2019), 569–575
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
Lazureanu C., Binzar T., “Symmetries and Properties of the Energy-Casimir Mapping in the Ball-Plate Problem”, Adv. Math. Phys., 2017, 5164602
A. A. Agrachev, “Topics in sub-Riemannian geometry”, Russian Math. Surveys, 71:6 (2016), 989–1019
A. Juhas, L. A. Novak, “Conflict set and waveform modelling for power amplifier design”, Math. Probl. Eng., 2015 (2015), 585962, 29 pp.
I. Yu. Beschastnyi, “The optimal rolling of a sphere, with twisting but without slipping”, Sb. Math., 205:2 (2014), 157–191
Proc. Steklov Inst. Math., 278 (2012), 218–232
A. P. Mashtakov, A. Yu. Popov, “Asymptotics of Maxwell Time in the Plate-Ball Problem”, Journal of Mathematical Sciences, 195:3 (2013), 336–368
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
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
A. P. Mashtakov, “Asymptotics of extremal curves in the ball rolling problem on the plane”, Journal of Mathematical Sciences, 199:6 (2014), 687–694
Citation in format AMSBIB
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