Circuito Exterior S/N, Ciudad Universitaria, Mexico City, 04510, Mexico
Depto. de Matemáticas, Facultad de Ciencias, UNAM
Bravo-Doddoli A., Montgomery R.
Geodesics in Jet Space
2022, vol. 27, no. 2, pp. 151-182
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.
Bravo-Doddoli A., García-Naranjo L. C.
The Dynamics of an Articulated $n$-trailer Vehicle
2015, vol. 20, no. 5, pp. 497-517
We derive the reduced equations of motion for an articulated $n$-trailer vehicle that moves under its own inertia on the plane. We show that the energy level surfaces in the reduced space are $(n + 1)$-tori and we classify the equilibria within them, determining their stability. A thorough description of the dynamics is given in the case $n = 1$.