The geometry of the plate-ball problem
type: Item
instance of: scholarly article
title: The geometry of the plate-ball problem
zbMATH Open document ID: 0809.70005
DOI: 10.1007/BF00375605
published in: Archive for Rational Mechanics and Analysis
publication date: 1995-04-04T00:00:00Z
review text: The author considers the plate-ball problem introduced recently by \textit{R. Brockett} and \textit{L. Dai} in an unpublished preprint. The author presents this as an optimal control problem on the five-dimensional Lie group \(G = {\mathbb{R}}^ 2 \times \text{SO}(3)\) described by the differential system \[ {dx_ 1\over dt} = u_ 1,\quad {dx_ 2 \over dt} = u_ 2, \quad {dR \over dt} = R \left( \begin{smallmatrix} 0 & 0 & -u_ 1 \\ 0 & 0 & -u_ 2\\ u_ 1 & u_ 2 & 0\end{smallmatrix} \right), \] where \(R\) is the rotation matrix in \(\text{SO}(3)\) and \(u_ 1\) and \(u_ 2\) are the control functions. One has to minimize \({1\over 2} \int_ 0 ^ T (u_ 1^ 2 + u_ 2^ 2) dt\) over all solution curves subject to the prescribed boundary condition in \(G\). The author shows that the problem is integrable in terms of elliptic functions. He shows also that the solution is related to the classical problem of the elastica described in a survey paper by \textit{T. Truesdell} [Bull. Am. Math. Soc. 9, 293-310 (1983; Zbl 0555.73030)].
reviewed by: Chandra Shekhar Sharma
Mathematics Subject Classification ID: 70E15
Mathematics Subject Classification ID: 70Q05
Mathematics Subject Classification ID: 22E70
zbMATH DE Number: 484802
zbMATH Keywords: integrable equations
zbMATH Keywords: problem of elastica
zbMATH Keywords: five-dimensional Lie group
zbMATH Keywords: elliptic functions
author: Velimir Jurdjevic
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 484802