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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
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.
Received: 05.10.2021 Accepted: 01.02.2022
Citation:
Alejandro Bravo-Doddoli, Richard Montgomery, “Geodesics in Jet Space”, Regul. Chaotic Dyn., 27:2 (2022), 151–182
Linking options:
https://www.mathnet.ru/eng/rcd1158 https://www.mathnet.ru/eng/rcd/v27/i2/p151
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Abstract page: | 104 | References: | 39 |
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