标题： 重力作用下山地斜坡上时间最小化导航的一般模型 显示英文标题

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

摘要： 在这项工作中，我们借助黎曼-芬斯勒几何解决了广义的松本“山之坡度”问题，并与泽梅洛导航问题建立了密切联系。 在重力作用下，分析了滑坡模型中时间最小化导航的一般情形，其中滑坡被建模为黎曼流形。 允许横向和纵向的重力附加量相对于运动方向同时在整个范围内变化，展示了横截面牵引和沿轨迹牵引对滑坡的影响。 通过背景黎曼度量的各向异性变形以及利用重新标定的引力风进行刚性平移，得到了最优导航的纯几何解，该解由属于广义$(\alpha, \beta)$-度量类的新芬斯勒度量给出。 建立了相关的强凸性条件并描述了时间测地线。 此外，通过几个二维例子，详细讨论并可视化了时间前沿的演化及在不同重力效应、引力风力和运动方向下的时间最小化轨迹行为。 显示英文摘要

摘要： 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.