Time geodesics on a slippery cross slope under gravitational wind

Abstract

In this work, we pose and solve the time-optimal navigation problem considered on a slippery mountain slope modeled by a Riemannian manifold of an arbitrary dimension, under the action of a cross gravitational wind. The impact of both lateral and longitudinal components of gravitational wind on the time geodesics is discussed. The varying along-gravity effect depends on traction in the presented model, whereas the cross-gravity additive is taken entirely in the equations of motion, for any direction and gravity force. We obtain the conditions for strong convexity and the purely geometric solution to the problem is given by a new Finsler metric, which belongs to the type of general $(\alpha ,\beta )$-metrics. The proposed model enables us to create a direct link between the Zermelo navigation problem and the slope-of-a-mountain problem under the action of a cross gravitational wind. Moreover, the behavior of the Finslerian indicatrices and time-minimizing trajectories in relation to the traction coefficient and gravitational wind force are explained and illustrated by a few examples in dimension two. This also compares the corresponding solutions on the slippery slopes under various cross- and along-gravity effects, including the classical Matsumoto’s slope-of-a-mountain problem and Zermelo’s navigation.