Time geodesics on a slippery cross slope under gravitational wind (Q6641219)

The authors address the time-optimal navigation problem on a slippery mountain slope modeled by a Riemannian manifold of any dimension, with the presence of a cross gravitational wind. They analyze how the lateral and longitudinal components of gravitational wind affect the time geodesics. The model considers the impact of traction on the along-gravity effect, while the cross-gravity additive is incorporated into the equations of motion for any direction and gravity force. The study establishes conditions for strong convexity and presents a geometric solution using a new Finsler metric of the general \((\alpha,\beta)\)-metrics type. This model establishes a connection between the Zermelo navigation problem and the mountain slope problem with a cross gravitational wind. The behaviour of Finslerian indicatrices and time-minimizing trajectories is explained through examples in two dimensions, comparing solutions under different cross- and along-gravity effects, including Matsumoto's slope-of-a-mountain problem and Zermelo's navigation.