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Mat. Sb., 2018, Volume 209, Number 5, Pages 74–119 (Mi msb8886)  

This article is cited in 1 scientific paper (total in 1 paper)

Liouville integrability of sub-Riemannian problems on Carnot groups of step 4 or greater

L. V. Lokutsievskiya, Yu. L. Sachkovb

a Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
b Ailamazyan Program Systems Institute of Russian Academy of Sciences, Yaroslavskaya obl., Pereslavskii raion, s. Ves'kovo

Abstract: One of the main approaches to investigating sub-Riemannian problems is Mitchell's theorem on nilpotent approximation, which reduces the analysis of a neighbourhood of a regular point to the analysis of the left-invariant sub-Riemannian problem on the corresponding Carnot group. Usually, the in-depth investigation of sub-Riemannian shortest paths is based on integrating the Hamiltonian system of Pontryagin's maximum principle explicitly. We give new formulae for sub-Riemannian geodesics on a Carnot group with growth vector $(2,3,5,6)$ and prove that left-invariant sub-Riemannian problems on free Carnot groups of step 4 or greater are Liouville nonintegrable.
Bibliography: 30 titles.

Keywords: sub-Riemannian geometry, Liouville integrability, Carnot groups, growth vector, separatrix splitting, Melnikov-Poincaré method.

Funding Agency Grant Number
Russian Science Foundation 14-50-00005
17-11-01387
Russian Academy of Sciences - Federal Agency for Scientific Organizations АААА-А17-117040610374-8
The work of L. V. Lokutsievskiy was supported by the Russian Science Foundation under grant no. 14-50-00005 at Steklov Mathematical Institute of Russian Academy of Sciences. The work of Yu. L. Sachkov in §§ 2 and 3 was supported by the Russian Science Foundation under grant no. 17-11-01387 and in §§ 7 and 11 it was carried out while fulfilling the state assignment (registered under no. AAAA-A17-117040610374-8) at the Program Systems Institute of the Russian Academy of Sciences. Sections 4, 5, 6 and 8 of this paper were written by L. V. Lokutsievskiy and sections 2, 3, 7 and 11 of this paper were written by Yu. L. Sachkov.

Author to whom correspondence should be addressed

DOI: https://doi.org/10.4213/sm8886

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English version:
Sbornik: Mathematics, 2018, 209:5, 672–713

Bibliographic databases:

UDC: 517.977
MSC: Primary 37J30, 53C17; Secondary 49J15
Received: 15.12.2016 and 14.02.2018

Citation: L. V. Lokutsievskiy, Yu. L. Sachkov, “Liouville integrability of sub-Riemannian problems on Carnot groups of step 4 or greater”, Mat. Sb., 209:5 (2018), 74–119; Sb. Math., 209:5 (2018), 672–713

Citation in format AMSBIB
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    This publication is cited in the following articles:
    1. Bolsinov A. Bao J., “A Note About Integrable Systems on Low-Dimensional Lie Groups and Lie Algebras”, Regul. Chaotic Dyn., 24:3 (2019), 266–280  crossref  isi
  • Математический сборник Sbornik: Mathematics (from 1967)
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