Sbornik: Mathematics
RUS  ENG    JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB  
General information
Latest issue
Forthcoming papers
Archive
Impact factor
Guidelines for authors
License agreement
Submit a manuscript

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS


Mat. Sb.:
Year:
Volume:
Issue:
Page:
Find




Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register

Sbornik: Mathematics, 2018, Volume 209, Issue 5, Pages 672–713
DOI: https://doi.org/10.1070/SM8886 (Mi sm8886) 		 
This article is cited in 18 scientific papers (total in 18 papers)

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
English version PDF (3251 kB) Citations (18)
References:
PDF
HTML
DOI: https://doi.org/10.1070/SM8886
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.
Received: 15.12.2016 and 14.02.2018
Russian version:
Matematicheskii Sbornik, 2018, Volume 209, Number 5, Pages 74–119
DOI: https://doi.org/10.4213/sm8886
Bibliographic databases:
Document Type: Article
UDC: 517.977
MSC: Primary 37J30, 53C17; Secondary 49J15
Language: English
Original paper language: Russian
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
\Bibitem{LokSac18}
\by L.~V.~Lokutsievskiy, Yu.~L.~Sachkov
\paper Liouville integrability of sub-Riemannian problems on Carnot groups of step 4 or greater
\jour Mat. Sb.
\yr 2018
\vol 209
\issue 5
\pages 74--119
\mathnet{http://mi.mathnet.ru/sm8886}
\crossref{https://doi.org/10.4213/sm8886}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=3795152}
\adsnasa{https://adsabs.harvard.edu/cgi-bin/bib_query?2018SbMat.209..672L}
\elib{https://elibrary.ru/item.asp?id=32823062}
\transl
\jour Sb. Math.
\yr 2018
\vol 209
\issue 5
\pages 672--713
\crossref{https://doi.org/10.1070/SM8886}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000439467500004}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85052021329}
Linking options:
  • https://www.mathnet.ru/eng/sm8886
  • https://doi.org/10.1070/SM8886
  • https://www.mathnet.ru/eng/sm/v209/i5/p74
    • This publication is cited in the following 18 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    		Математический сборник Sbornik: Mathematics
    Statistics & downloads:
    Abstract page:722
    Russian version PDF:69
    English version PDF:20
    References:65
    First page:32
     
      Contact us:
    		  Terms of Use  Registration to the website  Logotypes © Steklov Mathematical Institute RAS, 2024