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A curve of fastest descent. The problem of finding such a curve, which was posed by G. Galilei [1], is as follows: Out of all planar curves connecting two given points and and lying in a common vertical plane ( below ) to find the one along which a material point moving from to solely under the influence of gravity, would arrive at within the shortest possible time. The problem can be reduced to finding a function that constitutes a minimum of the functional

where and are the abscissas of points and . The brachistochrone is a cycloid with a horizontal base and with its cusp at the point .


[1] G. Galilei, "Unterredungen und mathematische Demonstrationen über zwei neue Wissenszweigen, die Mechanik und die Fallgesetze betreffend" , W. Engelmann , Leipzig (1891) (Translated from Italian and Greek)
[2] M.A. Lavrent'ev, L.A. Lyusternik, "A course in variational calculus" , Moscow-Leningrad (1950) (In Russian)


The brachistochrone problem is usually ascribed to Johann Bernoulli, cf. [a1], pp. 350 ff. A treatment can be found in most textbooks on the calculus of variations, cf. e.g. [a2].


[a1] W.W. Rouse Ball, "A short account of the history of mathematics" , Dover, reprint (1960) pp. 123–125
[a2] R. Weinstock, "Calculus of variations" , Dover, reprint (1974)
How to Cite This Entry:
Brachistochrone. L.P. Kuptsov (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098