Semi-basic 1-forms and Helmholtz conditions for the inverse problem of the calculus of variations

We use Fr\"olicher-Nijenhuis theory to obtain global Helmholtz conditions, expressed in terms of a semi-basic 1-form, that characterize when a semispray is locally Lagrangian. We also discuss the relation between these Helmholtz conditions and their classic formulation written using a multiplier matrix. When the semi-basic 1-form is 1-homogeneous (0-homogeneous) we show that two (one) of the Helmholtz conditions are consequences of the other ones. These two special cases correspond to two inverse problems in the calculus of variation: Finsler metrizability for a spray, and projective metrizability for a spray

Geometric properties of electromagnetic waves

This work studies geometrical properties of electromagnetic wave propagation. The work starts by studying geometrical properties of electromagnetic Gaussian beams in inhomogeneous anisotropic media. These are asymptotical solutions to Maxwell's equations that have a very characteristic feature. Namely, at each time instant the entire energy of the solution is concentrated around one point in space. When time moves forward, a Gaussian beam propagates along a curve. In recent work by A. P. Kachalov, Gaussian beams have been studied from a geometrical point of view. Under suitable conditions on the media, Gaussian beams propagate along geodesics. Furthermore, the shape of a Gaussian beam is determined by a complex tensor Riccati equation. The first paper of this dissertation provides a partial classification of media where Gaussian beams geometrize. The second paper shows that the real part of a solution to the aforementioned Riccati equation is essentially the shape operator for the phase front for the Gaussian beam. An important phenomena for electromagnetic Gaussian beams is that their propagation depend on their polarization. The last paper studies this phenomena from a very general point of view in arbitrary media. It also studies a connection between contact geometry and electromagnetism.reviewe