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Regular and Chaotic Dynamics, 2022, Volume 27, Issue 2, Pages 151–182
DOI: https://doi.org/10.1134/S1560354722020034
(Mi rcd1158)
 

This article is cited in 2 scientific papers (total in 2 papers)

Alexey Borisov Memorial Volume

Geodesics in Jet Space

Alejandro Bravo-Doddoli, Richard Montgomery

Dept. of Mathematics, UCSC, 1156 High Street, 95064 Santa Cruz, CA
Citations (2)
References:
Abstract: The space $J^k$ of $k$-jets of a real function of one real variable $x$ admits the structure of Carnot group type. As such, $J^k$ admits a submetry (sub-Riemannian submersion) onto the Euclidean plane. Horizontal lifts of Euclidean lines (which are the left translates of horizontal one-parameter subgroups) are thus globally minimizing geodesics on $J^k$.
All $J^k$-geodesics, minimizing or not, are constructed from degree $k$ polynomials in $x$ according to [7–9], reviewed here. The constant polynomials correspond to the horizontal lifts of lines. Which other polynomials yield globally minimizers and what do these minimizers look like? We give a partial answer. Our methods include constructing an intermediate three-dimensional “magnetic” sub-Riemannian space lying between the jet space and the plane, solving a Hamilton – Jacobi (eikonal) equations on this space, and analyzing period asymptotics associated to period degenerations arising from two-parameter families of these polynomials. Along the way, we conjecture the independence of the cut time of any geodesic on jet space from the starting location on that geodesic.
Keywords: Carnot group, Jet space, minimizing geodesic, integrable system, Goursat distribution, sub-Riemannian geometry, Hamilton – Jacobi, period asymptotics.
Funding agency Grant number
CONACYT - Consejo Nacional de Ciencia y Tecnología CVU 619610
This paper was developed with the support of the scholarship (CVU 619610) from “Consejo de Ciencia y Tecnologia” (CONACYT).
Received: 05.10.2021
Accepted: 01.02.2022
Bibliographic databases:
Document Type: Article
Language: English
Citation: Alejandro Bravo-Doddoli, Richard Montgomery, “Geodesics in Jet Space”, Regul. Chaotic Dyn., 27:2 (2022), 151–182
Citation in format AMSBIB
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\by Alejandro Bravo-Doddoli, Richard Montgomery
\paper Geodesics in Jet Space
\jour Regul. Chaotic Dyn.
\yr 2022
\vol 27
\issue 2
\pages 151--182
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  • https://www.mathnet.ru/eng/rcd1158
  • https://www.mathnet.ru/eng/rcd/v27/i2/p151
  • This publication is cited in the following 2 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
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    Abstract page:101
    References:38
     
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