arXiv subjects

Alejandro Bravo-Doddoli

Publications and source records attributed to Alejandro Bravo-Doddoli.

11 recordsLinked to original sources

arXiv2025

Metric Lines in the Space of Curves

This paper investigates sub-Riemannian geodesics within the jet space of curves. We establish the existence of two distinct families of metric lines, that is, globally minimizing geodesics, in the $2$-jet space of plane curves. This result provides an initial contribution toward the broader classification of metric lines in jet spaces. Additionally, we present precise criteria, which characterize when a sub-Riemannian geodesic in the $2$-jet space of plane curves can be identified as a metric line.

math.DG↗

arXiv2025

Integrable sub-Riemannian geodesic flows on the special orthogonal group

We analyse the geometry of the rubber-rolling distribution on the special orthogonal group and show that almost all the normal geodesics of any right-invariant sub-Riemannian metric defined on this distribution are completely integrable. Our argument is an adaptation of the method used to establish integrability of the Riemannian metric arising from the $n$-dimensional rigid body: namely, by exhibiting a Lax pair and bi-Hamiltonian structure for the reduced equations of motion.

math.DG↗

arXiv2025

Metric Lines in Engel-type Groups

In the framework of sub-Riemannian Manifolds, a relevant question is: what are the \enquote{metric lines} (i.e., the isometric embedding of the real line)? This article presents a conjecture classifying the metric lines in Carnot groups and takes the first steps in answering this question for \enquote{arbitrary rank} Carnot groups. We classify the metric lines of the Engel-type groups $\Eng(n)$ (Theorem 1.2), whose sub-Riemannian structure is defined on a non-integrable distribution of rank $n+1$. Our approach is a new method, called the sequence method, which we began to develop to study metric lines in the jet space.

math.DG↗

arXiv2023

Sympletic reduction of the sub-Riemannian geodesic flow for metabelian nilpotent groups

We consider nilpotent Lie groups for which the derived subgroup is abelian. We equip them with subRiemannian metrics and we study the normal Hamiltonian flow on the cotangent bundle. We show a correspondence between normal trajectories and polynomial Hamiltonians in some euclidean space. We use the aforementioned correspondence to give a criterion for the integrability of the normal Hamiltonian flow. As an immediate consequence, we show that in Engel-type groups the flow of the normal Hamiltonian is integrable. For Carnot groups that are semidirect products of two abelian groups, we give a set of conditions that normal trajectories must fulfill to be globally length-minimizing. Our results are based on a symplectic reduction procedure.

math.DG↗

arXiv2023

Metric lines in Jet Space

Given a sub-Riemannian manifold, a relevant question is: what are the metric lines (isometric embedding of the real line)? The space of $k$-jets of a real function of one real variable $x$, denoted by $J^k(\mathbb{R},\mathbb{R})$, admits the structure of a Carnot group, as every Carnot group $J^k(\mathbb{R},\mathbb{R})$ is a sub-Riemannian Manifold. This work is devoted to provide a partial result about the classification of the metric lines in $J^k(\mathbb{R},\mathbb{R})$. The method to prove the main Theorems is to use an intermediate $3$-dimensional sub-Riemannian space $\mathbb{R}^{3}_F$ lying between the group $J^k(\mathbb{R},\mathbb{R})$ and the Euclidean space $\mathbb{R}^{2} \simeq J^k(\mathbb{R},\mathbb{R}) / [J^k(\mathbb{R},\mathbb{R}),J^k(\mathbb{R},\mathbb{R})]$.

math.OC↗

arXiv2023

Chaotic subRiemannian geodesic flow in $J^2(\mathbb{R}^2,\mathbb{R})$

The space of $2$-jets of a real function of two real variables, denoted by $J^2(\mathbb{R}^2,\mathbb{R})$, admits the structure of a metabelian Carnot group, so $J^2(\mathbb{R}^2,\mathbb{R})$ has a normal abelian sub-group $\mathbb{A}$. As any sub-Riemannian manifold, $J^2(\mathbb{R}^2,\mathbb{R})$ has an associated Hamiltonian geodesic flow. The Hamiltonian action of $\mathbb{A}$ on $T^*J^2(\mathbb{R}^2,\mathbb{R})$ yields the reduced Hamiltonian $H_μ$ on $T^*\mathcal{H} \simeq T^*(J^2(\mathbb{R}^2,\mathbb{R})/\mathbb{A})$, where $H_μ$ is a two-dimensional Euclidean space. The paper is devoted to proving that reduced Hamiltonian $H_μ$ is non-integrable by meromorphic functions for some values of $μ$. This result suggests the sub-Riemannian geodesic flow on $J^{2}(\mathbb{R}^2,\mathbb{R})$ is not meromorphically integrable.

math.DS↗

arXiv2022

Geodesics in Jet Space

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-Peréz, 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.

math.OC↗

arXiv2022

No Periodic Geodesics in Jet Space

The $J^k$ space of $k$-jets of a real function of one real variable $x$ admits the structure of a sub-Riemannian manifold, which then has an associated Hamiltonian geodesic flow, and it is integrable. As in any Hamiltonian flow, a natural question is the existence of periodic solutions. Does $J^k$ have periodic geodesics? This study will find the action-angle coordinates in $T^*J^k$ for the geodesic flow and demonstrate that geodesics in $J^k$ are never periodic.

math.DS↗

arXiv2022

No Periodic normal Geodesics in $J^k(\mathbb{R},\mathbb{R}^n)$

The space of $k$-jets of $n$ real function of one real variable $x$ admits the structure of a Carnot group, which then has an associated Hamiltonian geodesic flow. As in any Hamiltonian flow, a natural question is the existence of periodic solutions. Does the space of $k$-jets have periodic geodesics? This study will demonstrate the integrability of subRiemannian geodesic flow, characterize and classify the subRiemannian geodesics in the space of $k$-jets, and show that they are never periodic.

math.DS↗

arXiv2020

Higher Elastica: Geodesics in the Jet Space

Carnot groups are subRiemannian manifolds. As such they admit geodesic flows, which are left-invariant Hamiltonian flows on their cotangent bundles. Some of these flows are integrable. Some are not. The space of k-jets for real-valued functions on the real line forms a Carnot group of dimension $k+2$. We show that its geodesic flow is integrable and that its geodesics generalize Euler's elastica, with the case $k=2$ corresponding to the elastica, as shown by Sachkov and Ardentov.

math.DS↗

arXiv2015

The dynamics of an articulated $n$-trailer vehicle

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 $n=1$.

math-ph↗