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Preprints, Working Papers, ... Year : 2021

Zermelo Navigation Problems on Surfaces of Revolution and Hamiltonian Dynamics

Problèmes de Navigation de Zermelo sur les Surfaces de Révolution et Dynamique Hamiltonienne


In this article, using the historical example from Carathéodory-Zermelo and a recent work describing the evolution of a passive tracer in a vortex, we present the geometric frame to analyze Zermelo navigation problems on surfaces of revolution, assuming the current invariant by symmetry of revolution. In this context, normal (polar) coordinates distinguish parallels and meridians and one will consider the case where the current is oriented along the parallels. The problem is set in the frame of time optimal control and the Maximum Principle allows to select minimizers among geodesics, solutions of an Hamiltonian dynamics. In the strong current domain, there exist both normal and abnormal geodesics, the later representing limit curves of the cone of admissible directions. We present the concepts of conjugate points in the normal and abnormal directions, associated to the singularity analysis of the central field defined by the Lagrangian manifold formed by geodesics curves with fixed initial conditions, in relation with Hamilton-Jacobi equation. This leads to a parameterization of the conjugate locus in the general case and conjugate points along abnormal directions are cusp points of the geodesics when crossing the limit of the weak current domain and are associated to non-continuity of the value function, due to bad accessibility propertie in the weak current domain. The dynamics is Liouville integrable but this dynamics is intricated due to noncompactness of the Liouville tori related to separatrices curves and interaction between parallels geodesics causing the existence of Reeb components. We present a generalized Morse-Reeb classification associated to an extended potential and this leads to a stratification of the geodesics set. Another complementary point of view is described which goes back to the historical example, using the parameterization of the geodesics by the heading angle of the ship, corresponding to the so-called Goh transformation in optimal control. This leads to a different stratification of the set of geodesics using Lie brackets computations and integration with Clairaut relation, interpreted as computing the control given by the derivative of the heading angle. The abnormal geodesics being seen as a projection with two branches of a determinantal variety. The final problem is to compute the cut locus and we introduce the first return mappings to equator and meridian to order the geodesics and to compute the separating locus formed by intersecting time minimizing geodesics. We apply our approach to analyse the historical example and a set of cases studies related in particular to Zermelo navigation on spheres of revolution and the generalized vortex case where the different techniques are used to analyze the dynamics of the geodesics and in fine to compute the optimal syntheses in the geodesically complete case.
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Dates and versions

hal-03209491 , version 1 (27-04-2021)
hal-03209491 , version 2 (07-05-2021)
hal-03209491 , version 3 (31-03-2022)
hal-03209491 , version 4 (07-03-2023)
hal-03209491 , version 5 (04-07-2023)


  • HAL Id : hal-03209491 , version 1


Bernard Bonnard, Olivier Cots, Boris Wembe. Zermelo Navigation Problems on Surfaces of Revolution and Hamiltonian Dynamics. 2021. ⟨hal-03209491v1⟩
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