Аннотация:
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)(3,6,14), the other is defined by two generatrices and growth vector (2,3,5,8)(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.
Ключевые слова:
sub-Riemannian geometry, Carnot group, Poincaré map, first integrals.
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).
Поступила в редакцию: 16.10.2016 Принята в печать: 20.11.2016
Образец цитирования:
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
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\paper Integrability and Nonintegrability of Sub-Riemannian Geodesic Flows on Carnot Groups
\jour Regul. Chaotic Dyn.
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\pages 759--774
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Образцы ссылок на эту страницу:
https://www.mathnet.ru/rus/rcd222
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Эта публикация цитируется в следующих 15 статьяx:
Ю. Л. Сачков, “Левоинвариантные задачи оптимального управления на группах Ли, интегрируемые в эллиптических функциях”, УМН, 78:1(469) (2023), 67–166; 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
Tijana Sukilovic, Srdjan Vukmirovic, “Riemannian and sub-Riemannian structures on a cotangent bundle of Heisenberg group”, Filomat, 37:25 (2023), 8481
Ю. Л. Сачков, “Субриманова сфера Картана”, Докл. РАН. Матем., информ., проц. упр., 507 (2022), 66–70; Yu. L. Sachkov, “Sub-Riemannian Cartan sphere”, Dokl. Math., 106:3 (2022), 462–466
Andrei Ardentov, Eero Hakavuori, “Cut time in the sub-Riemannian problem on the Cartan group”, ESAIM: COCV, 28 (2022), 12
Podobryaev A.V., “Casimir Functions of Free Nilpotent Lie Groups of Steps 3 and 4”, J. Dyn. Control Syst., 27:4 (2021), 625–644
Yu. L. Sachkov, “Conjugate time in the sub-Riemannian problem on the Cartan group”, J. Dyn. Control Syst., 27:4 (2021), 709–751
Yuri L. Sachkov, “Periodic Controls in Step 2 Strictly Convex Sub-Finsler Problems”, Regul. Chaotic Dyn., 25:1 (2020), 33–39
A. P. Mashtakov, A. Yu. Popov, “Asymptotics of Extremal Controls in the Sub-Riemannian Problem on the Group of Motions of Euclidean Space”, Rus. J. Nonlin. Dyn., 16:1 (2020), 195–208
Ю. Л. Сачков, “Коприсоединенные орбиты и задачи быстродействия
для двухступенных свободных нильпотентных групп Ли”, Матем. заметки, 108:6 (2020), 899–910; Yu. L. Sachkov, “Coadjoint Orbits and Time-Optimal Problems for Step-2 Free Nilpotent Lie Groups”, Math. Notes, 108:6 (2020), 867–876
Alexey Bolsinov, Jinrong Bao, “A Note about Integrable Systems on Low-dimensional Lie Groups and Lie Algebras”, Regul. Chaotic Dyn., 24:3 (2019), 266–280
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
Alexey P. Mashtakov, A. 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
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
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