I am interested in the role of symmetry and geometry in the dynamics of mechanical systems. I work on non-integrable distributions from the point of view of the sub-Riemannian geometry and the non-holonimcs mechanics. In particular, I focus on sub-Riemannian geodesics in Carnot groups.

Unas voces le hablan al profesor Barajas cuando estaba estudiando en San Idelfonso la preparatoria. "No hay sino anhelos, Barajas. Lo demás no existe. Por lo menos no existe vitalmente. La realidad de que habla la ciencia es una realidad pensada. Realidad solamente la tienen las cosas cuando en ellas se prende nuestro deseo o nuestra nostalgia. La India ha sabido esto mejor que nadie; por eso Buda hace de la sed la sustancia del mundo. Nos hacemos la ilusión de que somos mercaderes, pero nuestra caravana salió al desierto sólo para sentir sed. Sed de vivir, sed de conocer, sed de morir".

Luis Garcia-Narajo and I studied the dynamics of an articulated n-trailer vehicle that moves under its own inertia. Such a system consists of a leading car, or truck, pulling n trailers, like a luggage carrier in the airport. The wheels on the car and each truck impose a nonholonomic constraint that forbids any motion of the given body in the direction perpendicular to its central axis. The space of configuration Q is an (n+3) dimensional manifold, which is endowed with a Lagrangian function given by the kinetic energy and a two-rank non-integrable distribution D. The Euclidean group SE(2) acts on Q, the lagrangian function, and the distribution D are invariant under the action, the quotient of Q by SE(2) is an n-torus T^n. We wrote the equations of motions on the reduced space T^n. If a = 0, the system has a new constant of motion, namely, the angular velocity of the leading car, and the dynamics changes substantially. The following videos show the different behaviors of the system. The center of mass is the red point, the car is green, and the trailers are blue.

(Case n=1) The reduced dynamics is integrable and has a singular energy level E_c: Energy E less than E_c, the reduce dynamics is periodic and its reconstruction are periodic (Periodic Orbit) or quasi-periodic (Quasi-periodic Orbit). Energy E equal E_c, the reduced dynamics is a homoclinic connections whose equilibrium point correspond to periodic solutions in the reconstructed dynamics ( Homoclinic Orbit). Energy E is bigger than E_c, the reduced dynamics is a heteroclinic connection, and its reconstructed dynamics is asymptotic to a periodic solution corresponding to the equilibrium points (Heteroclinic Orbit).

(Case n=2) The dynamics has two singular energy levels E1_c and E2_c, see (Orbit with energy E1_c) and ( Orbit with energy E2_c). A solution with energy bigger than E1_c and E2_c is here (Orbit with energy bigger than E1_c and E2_c ).

(Case n=1) The following videos show a well-known phenomenon; the stable relative equilibrium points correspond to configurations of the system when the center of mass is located forward in the direction of the movement. In the following videos, the system starts with small angular velocity and its reach its maximum before to be asymptotically to 0, smaller maximum angular velocity (Periodic Orbit), medium maximum angular velocity (Periodic Orbit) and bigger maximum angular velocity (Periodic Orbit).