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The brachistochrone problem with frictional forces

Published online by Cambridge University Press:  15 August 2002

Roberto Giambò  and
Fabio Giannoni
Roberto Giambò
Affiliation:
Dipartimento di Matematica “Ulisse Dini”, Università di Firenze, Italy; giambo@udini.math.unifi.it.
Fabio Giannoni
Affiliation:
Dipartimento di Matematica e Fisica, Università di Camerino, Italy; giannoni@campus.unicam.it.
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Abstract

In this paper we show the existence of the solution for the classical brachistochrone problem under the action of a conservative field in presence of frictional forces. Assuming that the frictional forces and the potential grow at most linearly, we prove the existence of a minimizer on the travel time between any two given points, whenever the initial velocity is great enough. We also prove the uniqueness of the minimizer whenever the given points are sufficiently close.

Keywords

Minimal travel timenon linear constraints.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 5 , 2000 , pp. 187 - 206
Copyright
© EDP Sciences, SMAI, 2000

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References

H. Brezis, Analyse fonctionnelle. Masson, Paris (1983).
Giannoni, F., Piccione, P. and Verderesi, J.A., An approach to the relativistic brachistochrone problem by sub-Riemannian geometry. J. Math. Phys . 38 (1997) 6367-6381. CrossRef
Giannoni, F. and Piccione, P., An existence theory for relativistic brachistochrones in stationary space-times. J. Math. Phys . 39 (1998) 6137-6152. CrossRef
Nash, J., The embedding problem for Riemannian manifolds. Ann. Math. 63 (1956) 20-63. CrossRef
L.A. Pars, An Introduction to the Calculus of Variations. Heinemann, London (1962).
Perlick, V., The brachistochrone problem in a stationary space-time. J. Math. Phys. 32 (1991) 3148-3157. CrossRef