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Regul. Chaotic Dyn., 2016, Volume 21, Issue 6, Pages 759–774 (Mi rcd222)  

This article is cited in 3 scientific papers (total in 3 papers)

Integrability and Nonintegrability of Sub-Riemannian Geodesic Flows on Carnot Groups

Ivan A. Bizyaeva, Alexey V. Borisovba, Alexander A. Kilina, Ivan S. Mamaevc

a Udmurt State University, ul. Universitetskaya 1, Izhevsk, 426034 Russia
b National Research Nuclear University MEPhI, Kashirskoe sh. 31, Moscow, 115409 Russia
c Izhevsk State Technical University, ul. Studencheskaya 7, Izhevsk, 426069 Russia

Abstract: This paper is concerned with two systems from sub-Riemannian geometry. One of them is defined by a Carnot group with three generatrices and growth vector $(3,6,14)$, the other is defined by two generatrices and growth vector $(2,3,5,8)$. Using a Poincaré map, the nonintegrability of the above systems in the general case is shown. In addition, particular cases are presented in which there exist additional first integrals.

Keywords: sub-Riemannian geometry, Carnot group, Poincaré map, first integrals

Funding Agency Grant Number
Ministry of Education and Science of the Russian Federation
Dynasty Foundation
Russian Foundation for Basic Research 14-01-00395-a
Russian Science Foundation 14-19-01303
The work of I. A. Bizyaev and A. A. Kilin was carried out within the framework of the state assignment for institutions of higher education. The work of I. A. Bizyaev was also partially supported by the Dynasty Foundation. The work of A. V. Borisov was supported by the RFBR grant No. 14-01-00395-a. The work of I. S. Mamaev (Section 3) was supported by the Russian Science Foundation (project No. 14-19-01303).


DOI: https://doi.org/10.1134/S1560354716060125

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Document Type: Article
MSC: 53C17, 37C10
Received: 16.10.2016
Language: English

Citation: Ivan A. Bizyaev, Alexey V. Borisov, Alexander A. Kilin, Ivan S. Mamaev, “Integrability and Nonintegrability of Sub-Riemannian Geodesic Flows on Carnot Groups”, Regul. Chaotic Dyn., 21:6 (2016), 759–774

Citation in format AMSBIB
\Bibitem{BizBorKil16}
\by Ivan A. Bizyaev, Alexey V. Borisov, Alexander A. Kilin, Ivan S. Mamaev
\paper Integrability and Nonintegrability of Sub-Riemannian Geodesic Flows on Carnot Groups
\jour Regul. Chaotic Dyn.
\yr 2016
\vol 21
\issue 6
\pages 759--774
\mathnet{http://mi.mathnet.ru/rcd222}
\crossref{https://doi.org/10.1134/S1560354716060125}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3583949}
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Evgeny V. Vetchanin, Alexander A. Kilin, Ivan S. Mamaev, “Control of the Motion of a Helical Body in a Fluid Using Rotors”, Regul. Chaotic Dyn., 21:7-8 (2016), 874–884  mathnet  crossref
    2. Boris S. Kruglikov, Andreas Vollmer, Georgios Lukes-Gerakopoulos, “On Integrability of Certain Rank 2 Sub-Riemannian Structures”, Regul. Chaotic Dyn., 22:5 (2017), 502–519  mathnet  crossref
    3. Alexey P. Mashtakov, Anton Yu. Popov, “Extremal Controls in the Sub-Riemannian Problem on the Group of Motions of Euclidean Space”, Regul. Chaotic Dyn., 22:8 (2017), 949–954  mathnet  crossref
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