  In the present work, we study the optimal control paths in the Zermelo
navigation problem from the geometric and differential equations point of view
rather than the optimal control point of view, where the latter has been
carried out in our recent work. Here, we obtain the precise form of the system
of ODE where the solutions are optimal trajectories of Zermelo's navigation
problem. Having a precise equation allows optimizing a cost function more
accurately and efficiently. The advantage of these equations is to approximate
optimal trajectories in the general case by the first-order approximation of
external fields $w$. The latter could be solved numerically since we have
retrieved simpler equations for these paths.
