Abstract
In this paper we use a variational approach, combining holonomic and nonholonomic constraints, to find an equation for sub-Riemannian geodesics on the orthogonal group. This approach is extrinsic in nature and makes the paper fully self-contained and possibly more accessible for a wide audience. The problem is formulated in the vector space of real square matrices, subject to two side conditions, and solved using a Langrange multiplier approach. The nonholonomic constraint corresponds to the requirement that the curves are tangent to a left-invariant distribution. This distribution is defined by the vector space that shows up in a Cartan decomposition of the Lie algebra associated to the orthogonal group.
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Acknowledgements
This work was initiated when the first and the third authors were visiting ICMAT in Madrid in November 2019. The hospitality of our hosts, David Martín de Diego and Leonardo Colombo, is greatly appreciated. This work was continued when the authors met in 2020, at the Institute for Mathematics in Würzburg, Germany.
The first author has been supported in part by the German Federal Ministry of Education and Research (BMBF-Projekt 05M20WWA: Verbundprojekt 05M2020 - DyCA).
The second author was partially supported by the project Pure Mathematics in Norway, funded by the Trond Mohn Foundation.
The third author acknowledges Fundação para a Ciência e a Tecnologia (FCT) and COMPETE 2020 program for the financial support to the project UIDB/00048/2020.
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Hüper, K., Markina, I., Silva Leite, F. (2021). An Extrinsic Approach to Sub-Riemannian Geodesics on the Orthogonal Group. In: Gonçalves, J.A., Braz-César, M., Coelho, J.P. (eds) CONTROLO 2020. CONTROLO 2020. Lecture Notes in Electrical Engineering, vol 695. Springer, Cham. https://doi.org/10.1007/978-3-030-58653-9_26
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DOI: https://doi.org/10.1007/978-3-030-58653-9_26
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