This paper aims at solving Zermelo’s navigation problem on conformally flat Riemannian manifolds admitting a ship’s variable self-speed, under the action of arbitrary winds including space and time dependence for both perturbation and ship’s speed. Our approach is a variational one under application of the Euler–Lagrange equations with reference to the initial studies of this problem. First of all, we distinguish the navigation cases in non-critical, i.e. weak or strong, and critical winds, which are then unified into an arbitrary wind. After having considered the second variation of a given functional, we obtain the conditions for both time-minimal, i.e. the typical solutions to Zermelo’s problem, and time-maximal extremals. The anomalous paths are also emphasized. Moreover, some classification results are presented with respect to the kinds of perturbation considered separately and under an arbitrary wind. This study is illustrated at its end by a two-dimensional example including a prolate ellipsoid in the presence of a rotational vector field, wherein the solution types are being compared.

本文的目的是在任意风的作用下（包括扰动和速度对空间和时间的依赖），解决准线平坦的黎曼流形上的Zermelo导航问题，该流形允许船舶具有可变的自速度。我们的方法是在应用Euler-Lagrange方程的基础上，对这一问题进行了初步研究的一种变通方法。首先，我们将导航情况区分为非关键性（即弱风或强风）和关键性风，然后将其合并为任意风。在考虑了给定函数的第二种变化之后，我们获得了最小时间（即Zermelo问题的典型解）和最大时间极值的条件。还强调了异常路径。此外，对于分别考虑在任意风下的摄动类型，给出了一些分类结果。这项研究在最后以一个二维示例说明，该示例包括在旋转矢量场存在下的长椭球体，其中对溶液类型进行了比较。