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A Global algorithm for geodesics

Part of:Numerical methods in calculus of variations and optimal controlBoundary value problemsGlobal differential geometry

Published online by Cambridge University Press:  09 April 2009

Lyle Noakes
Lyle Noakes
Affiliation:
Department of Mathematics, The University of Western Australia, Nedlands WA 6907, Australia email: lyle@maths.uwa.edu.au
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Abstract

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The problem of finding a george joinning 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 expectually 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 to elementary and very carefully stated. with a view to possible generalizations latter on we have in mind the much larger class of interesting problems arising in optimal control especially from mechanical engineering.

MSC classification

Secondary:34B15: Nonlinear boundary value problems49M05: Methods based on necessary conditions53C22: Geodesics

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 65 , Issue 1 , August 1998, pp. 37 - 50
Copyright
Copyright © Australian Mathematical Society 1998

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