Research

Research in sub-Riemannian geometry, geometric mechanics, and geometric control theory.

My research lies at the intersection of sub-Riemannian geometry, geometric mechanics, and geometric control theory. My main interests include the classification of globally minimizing geodesics, the geometry and dynamics of geodesic flows on Lie groups, the integrability of Hamiltonian systems, and the global dynamics of nonholonomic mechanical systems, particularly the transition from integrable to chaotic behavior.

Metric Lines and the Sequence Method

One of my main research interests is the classification of metric lines, namely, geodesics that minimize distance globally between any two of their points. Understanding these curves is a fundamental problem in sub-Riemannian geometry, where the presence of nonholonomic constraints makes the global behavior of geodesics particularly challenging to analyze.

The state of the art in establishing the global minimality of sub-Riemannian geodesics has traditionally involved techniques such as optimal synthesis and Hamilton–Jacobi theory. These approaches often require explicit integration of the geodesic equations or the construction of suitable calibrations. In contrast, I developed the sequence method, which establishes global minimality through compactness arguments and the geometric properties of minimizing sequences.

I have applied the sequence method to investigate metric lines in several Carnot groups, including jet spaces and Engel-type groups. These applications illustrate how the method can be used to study global minimality without explicitly integrating the geodesic equations.

Integrability and Reduction of Geodesic Flows

Another direction of my research focuses on the integrability of sub-Riemannian geodesic flows on Lie groups and the use of reduction techniques to investigate their dynamics.

In joint work with P. Arathoon and A. M. Bloch, we established the Liouville integrability of a sub-Riemannian geodesic flow on the orthogonal group using Poisson reduction. A key ingredient of our approach is the construction of first integrals for the sub-Riemannian flow by taking suitable limits of first integrals of the corresponding Riemannian geodesic flows.

In a related project with E. Le Donne and N. Paddeu, we developed a framework for the symplectic reduction of sub-Riemannian geodesic flows on metabelian nilpotent Lie groups, using normal abelian subgroups. As an application of this framework, we established the Liouville integrability of the sub-Riemannian geodesic flow on Engel-type Carnot groups.

Nonholonomic Mechanics and Billiards

My research in geometric mechanics focuses on nonholonomic systems, whose motion is subject to non-integrable velocity constraints. A central goal of my research is to understand their global dynamics, particularly the mechanisms underlying the transition from integrable to chaotic behavior.

In joint work with A. M. Bloch and W. Clark, we established the integrability of a knife-edge billiard moving inside a circular domain. This system combines nonholonomic motion with elastic impacts at the boundary. Our analysis reveals the geometric structure of the impact dynamics and establishes the existence of a family of generalized caustics.

In earlier joint work with L. C. García-Naranjo, we investigated the dynamics of an articulated n-trailer vehicle, a classical nonholonomic mechanical system, focusing on the qualitative behavior of its trajectories.

For a detailed discussion, including figures and numerical simulations, see my n-Trailer Vehicle Project page.

Related publications:

Optimal Synthesis and Geometric Control

Another direction of my research focuses on the application of geometric optimal control methods to determine cut times, cut loci, and globally minimizing geodesics.

Together with my undergraduate student Marcus T. C. Lau, we investigated the Riemannian geometry of the special linear group $\mathbb{SL}(2,\mathbb{R})$. We obtained a geometric characterization of the cut locus, showing that it consists of symmetric negative-definite matrices. We also derived an explicit formula for the injectivity radius.

These results contribute to understanding the global geometry of Riemannian geodesics on Lie groups.

Related publication: