Zermelo's problem, Finsler spacetimes and applications

Résumé

This thesis is a collection of four research articles and features a series of results, linked to Zermelo's problem, Finsler spacetimes and applications to wave propagation, that are relevant both from a theoretical and practical perspective. From a theoretical point of view, Articles I and II study the time-minimizing properties of lightlike geodesics in Finsler spacetimes and present a geometric framework where they can be interpreted as wave trajectories—or, more generally, trajectories of any physical phenomenon that satisfies Huygens' principle—in a general context: the wave can be anisotropic (direction-dependent) and rheonomic (time-dependent) in any dimension and with an arbitrary initial wavefront. These trajectories generalize the solutions to the classical Zermelo's problem, which seeks the fastest trajectory between two fixed points for a moving object that travels in the presence of a current. Within this setting, the wave velocities yield a Finsler metric on the space (and the corresponding cone structure in the spacetime) and the computation of the wavefront is reduced to solving an ODE system—the geodesic equations of a Lorentz-Finsler metric. Article IV completes this framework with the generalized Snell's and reflection laws, which provide the refracted and reflected trajectories when the wave crosses the interface between two different media. These laws generalize the corresponding classical ones when the wave propagation is anisotropic. From a practical point of view, this thesis focuses on wildfire spread modeling. Article II proves that the aforementioned ODE system, in the simplest case when the fire growth is assumed to be elliptical, is equivalent to the PDE system used by current fire growth simulators to compute the firefront. In Article III we develop a non-elliptical model, constructing a specific Finsler metric that accounts for the anisotropies generated by the wind and the slope—the main sources of anisotropy. On the one hand, the slope effect is modeled by an inverse Matsumoto metric, which favors the upward direction—the fire moves faster upwards than downwards. In the presence of wind, on the other hand, the fire spread takes the approximate shape of a double semi-ellipse, which constitutes a good experimental fitting. This model is still in an early development stage and must be tested experimentally. Anyway, its true value lies not in the metric itself—which can be easily modified to fit any other strongly convex shape—but in the novel use of Finsler geometry, which simplifies the computation of the firefront and allows us to effectively overcome the elliptical constraint.