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
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.
sub-Riemannian geometry, Liouville integrability, Carnot groups, growth vector, separatrix splitting, Melnikov-Poincaré method.
|Russian Science Foundation
|Russian Academy of Sciences - Federal Agency for Scientific Organizations
|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.
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Sbornik: Mathematics, 2018, 209:5, 672–713
MSC: Primary 37J30, 53C17; Secondary 49J15
Received: 15.12.2016 and 14.02.2018
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
\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.
\jour Sb. Math.
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