Doklady Mathematics

, Volume 95, Issue 3, pp 211–213 | Cite as

Liouville nonintegrability of sub-Riemannian problems on free Carnot groups of step 4

Mathematics

Abstract

One of the main approaches to the study of the Carnot–Carathéodory metrics is the Mitchell–Gromov nilpotent approximation theorem, which reduces the consideration of a neighborhood of a regular point to the study of the left-invariant sub-Riemannian problem on the corresponding Carnot group. A detailed analysis of sub-Riemannian extremals is usually based on the explicit integration of the Hamiltonian system of Pontryagin’s maximum principle. In this paper, the Liouville nonintegrability of this system for left-invariant sub-Riemannian problems on free Carnot groups of step 4 and higher is proved.

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Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia
  2. 2.Mechanics and Mathematics FacultyMoscow State UniversityMoscowRussia
  3. 3.Ailamazyan Program Systems InstituteRussian Academy of Sciencess. Ves’kovoRussia
  4. 4.RUDN UniversityMoscowRussia

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