Brachistochrone Problem

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Problem

Let a point $A$ be joined by a wire to a lower point $B$.

Let the wire be allowed to be bent into whatever shape is required.

Let a bead be released at $A$ to slide down without friction to $B$.


What is the shape of the wire so that the bead takes least time to descend from $A$ to $B$?


Also see


Historical Note

The Brachistochrone Problem was raised by Johann Bernoulli to the readers of Acta Eruditorum in June $1696$.

Isaac Newton interpreted the problem as a direct challenge to his abilities, and (despite being out of practice) solved the problem in the evening before going to bed.

He published it anonymously, but Bernoulli recognised whose solution it was, and commented:

I recognise the lion by his print.

Bernoulli published the solution in the Acta Eruditorum in May $1697$, along with solutions by Jacob Bernoulli and Gottfried Wilhelm von Leibniz.

With justice we admire Huygens because he first discovered that a heavy particle slides down to the bottom of a cycloid in the same time, no matter where it starts. But you will be petrified with astonishment when I say that this very same cycloid, the tautochrone of Huygens, is also the brachistochrone we are seeking.
-- Johann Bernoulli


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