Regular and Chaotic Dynamics

, Volume 21, Issue 6, pp 759–774 | Cite as

Integrability and nonintegrability of sub-Riemannian geodesic flows on Carnot groups

  • Ivan A. Bizyaev
  • Alexey V. Borisov
  • Alexander A. Kilin
  • Ivan S. Mamaev
On the 70th Birthday of Nikolai N. Nekhoroshev Special Memorial Issue. Part 1

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 

MSC2010 numbers

53C17 37C10 

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Copyright information

© Pleiades Publishing, Ltd. 2016

Authors and Affiliations

  • Ivan A. Bizyaev
    • 1
  • Alexey V. Borisov
    • 1
    • 2
  • Alexander A. Kilin
    • 1
  • Ivan S. Mamaev
    • 3
  1. 1.Udmurt State UniversityIzhevskRussia
  2. 2.National Research Nuclear University “MEPhI”MoscowRussia
  3. 3.Izhevsk State Technical UniversityIzhevskRussia

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