|XXVIII Workshop on Geometric Methods in Physics||28.06-04.07.2009|
Geometrical particles on null curves in Lorentzian space forms
We study geometrical particles modeled by actions whose Lagrangians are arbitrary functions on the curvature of null paths in $(d+1)$-dimensions Lorentzian space forms. These models correspond with relativistic particles interacting with external gauge and gravitational fields. The geometrical appproach is based on the description of the particle worldline by its geometrical invariants (the curvatures) instead of its position vector and its higher derivatives. We obtain first integrals of the Euler-Lagrange equation by using geometrical methods involving the search for killing vector fields along critical curves of the action. Some particular cases where Lagrangian density depends quadratically on Cartan curvature are analyzed and it is shown that the mechanical systems are governed by stationary Korteweg-De Vries and Henon-Heiles systems.