Amirkabir University of Technology AUT Journal of Mathematics and Computing 2783-2449 3 1 2022 02 01 On the geometry of Zermelo’s optimal control trajectories 1 10 4542 10.22060/ajmc.2021.20459.1066 EN Zohreh Fathi Faculty of Mathematics and Computer Sciences, Amirkabir University of Technology (Tehran Polytechnic), 424 Hafez Ave. 15914 Tehran, Iran Behroz Bidabad Faculty of Mathematics and Computer Sciences, Amirkabir University of Technology (Tehran Polytechnic), 424 Hafez Ave. 15914 Tehran, Iran Institut de Math´ematique de Toulouse, Universit´e Paul Sabatier, F-31062 Toulouse, France 0000-0003-3993-4268 Journal Article 2021 08 24 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. https://ajmc.aut.ac.ir/article_4542_7c33e57e3dbd8a52940fa1a963aa4a4a.pdf