A Global Algorithm for Geodesics

The problem of finding a geodesic joining given points x0; x1 in a connected complete Riemannian manifold requires much more effort than determining a geodesic from initial data. Boundary value problems of this type are sometimes solved using shooting methods, which work best when good initial guesses are available, especially when x0; x1 are nearby. Galerkin methods have their drawbacks too. The situation is much more difficult with general variational problems, which is why we focus on the Riemannian case. Our global algorithm is very simple to implement, and works well in practice, with no need for an initial guess. The proof of convergence is elementary and very carefully stated, with a view to possible generalizations later on. We have in mind the much larger class of interesting problems arising in optimal control especially from mechanical engineering. 1991 Mathematics subject classification (Amer. Math. Soc.): primary 34B15, 49M05; secondary 53C22.