| Autor: |
Bravo-Doddoli, Alejandro, Montgomery, Richard |
| Rok vydání: |
2021 |
| Předmět: |
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| Zdroj: |
Regular and Chaotic Dynamics, 2022 |
| Druh dokumentu: |
Working Paper |
| DOI: |
10.1134/S1560354722020034 |
| Popis: |
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 (\sR 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 Anzaldo-Meneses and Monroy-Per\'ez, 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. |
| Databáze: |
arXiv |
| Externí odkaz: |
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