Amirkabir University of TechnologyAUT Journal of Mathematics and Computing2783-24493120220201On the geometry of Zermelo's optimal control trajectories110454210.22060/ajmc.2021.20459.1066ENZohrehFathiFaculty of Mathematics Amirkabir University of TechnologyBehrozBidabadDepartment of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic)0000-0003-3993-4268Journal Article20210824In 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.<br />Having a precise equation allows optimizing a cost function more accurately and efficiently.<br />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.https://ajmc.aut.ac.ir/article_4542_9c3b0722a6e3294b44ca6b90179f05a7.pdf