A general model for time-minimizing navigation on a mountain slope under gravity

Abstract

In this work, we solve the generalized Matsumoto's slope-of-a-mountain problem by means of Riemann-Finsler geometry, making close links with the Zermelo navigation problem. The time-minimizing navigation under gravity is analyzed in the general model of a slippery mountain slope being a Riemannian manifold. Both the transverse and longitudinal gravity-additives with respect to the direction of motion are admitted to vary simultaneously in the full ranges, showing the impact of cross- and along-traction on the slippery slope. By the anisotropic deformation of the background Riemannian metric and rigid translation with the use of the rescaled gravitational wind, we obtain the purely geometric solution for optimal navigation, which is given by a new Finsler metric belonging to the class of general $(\alpha, \beta)$-metrics. The related strong convexity conditions are established and time geodesics are described. Moreover, the evolution of time fronts and the behavior of time-minimizing trajectories in relation to various gravity effects on the slippery slope, gravitational wind force and direction of motion are thoroughly discussed and visualized by several two-dimensional examples.