Manuel Barros

Departamento de Geometría y Topología, Universidad de Granada

Geometry and Dynamics of relativistic particles and strings
The chief point in the conventional approach to search Lagrangians that describe spinning particles is based in an extension of the original space-time by extra variables that provide the required new degrees of freedom.

However, in recent years, a new approach appeared in the literature. This consists in describing the particle systems by Lagrangians that, being formulated in the original space-time (they are intrinsic), in return for they depend on higher derivatives. An obvious argument involving the Poincaré invariance implies that the admissible Lagrangian densities must depend on the curvatures of the particles trajectories in the original background gravitational field.
The following motivations, and more, lie in the core of this new approach:

These models seem to continue realizing an old idea of modelling spinning particles without introducing additional spin variables
These models imitate the Polyakov one for bosonic strings. Therefore, these extend the free particle model in the same way that the Polyakov system extend the classical Nambu-Goto model of strings.
These models are also connected with problems describing planar phenomena such as the high-temperature superconductivity and the problem of Landau-Hall.
These models could provide solutions for particles with arbitrary spin (the anyons) and also theories of particles with maximal proper acceleration.

At the time being, these models have independent interest and had taken on a life of its own as can be checked from its extensed and nice literature concerning with.

In these lectures, we will deal with these kind of particle and string models, paying special attention to their field equations. We will get many solutions from different geometrical arguments and, many times, algorithms to obtain the complete moduli spaces of solutions.