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Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2023, Volume 321, Pages 128–155
DOI: https://doi.org/10.4213/tm4301
(Mi tm4301)
 

Lie Algebras and Integrable Systems: Elastic Curves and Rolling Geodesics

V. Jurdjevic

Department of Mathematics, University of Toronto, Toronto, ON, M5S 2E4, Canada
References:
Abstract: This paper follows a long-standing fascination in the relevance of Lie algebras and Lie groups for problems of applied mathematics. It originates with the discovery that the mathematical formalism initiated by G. Kirchhoff to model the equilibrium configurations of an elastic rod can be extended to the isometry groups of certain Riemannian manifolds through control theoretic insights and the Maximum Principle, giving rise to a large class of Hamiltonian systems that link geometry with physics in novel ways. This paper focuses on the relations between the Kirchhoff-like affine–quadratic problem and the rolling geodesic problem associated with the rollings of homogeneous manifolds $G/K$, equipped with a $G$-invariant metric, on their tangent spaces. We will show that there is a remarkable connection between these two problems manifested through a common isospectral curve in the Lie algebra $\mathfrak g$ of $G$. In the process we will reveal the significance of curvature for the theory of elastic curves.
Received: February 23, 2022
Revised: July 5, 2022
Accepted: January 9, 2023
English version:
Proceedings of the Steklov Institute of Mathematics, 2023, Volume 321, Pages 117–142
DOI: https://doi.org/10.1134/S0081543823020098
Bibliographic databases:
Document Type: Article
UDC: 514.852+517.977
Language: Russian
Citation: V. Jurdjevic, “Lie Algebras and Integrable Systems: Elastic Curves and Rolling Geodesics”, Optimal Control and Dynamical Systems, Collected papers. On the occasion of the 95th birthday of Academician Revaz Valerianovich Gamkrelidze, Trudy Mat. Inst. Steklova, 321, Steklov Math. Inst., Moscow, 2023, 128–155; Proc. Steklov Inst. Math., 321 (2023), 117–142
Citation in format AMSBIB
\Bibitem{Jur23}
\by V.~Jurdjevic
\paper Lie Algebras and Integrable Systems: Elastic Curves and Rolling Geodesics
\inbook Optimal Control and Dynamical Systems
\bookinfo Collected papers. On the occasion of the 95th birthday of Academician Revaz Valerianovich Gamkrelidze
\serial Trudy Mat. Inst. Steklova
\yr 2023
\vol 321
\pages 128--155
\publ Steklov Math. Inst.
\publaddr Moscow
\mathnet{http://mi.mathnet.ru/tm4301}
\crossref{https://doi.org/10.4213/tm4301}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=4643637}
\transl
\jour Proc. Steklov Inst. Math.
\yr 2023
\vol 321
\pages 117--142
\crossref{https://doi.org/10.1134/S0081543823020098}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85170834142}
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    Труды Математического института имени В. А. Стеклова Proceedings of the Steklov Institute of Mathematics
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