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Brachistochrone Part I
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Brachistochrone Part IV

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Equations for the Brachistochrone

Brachistochrone Part II
 The Equations and History

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Equations of the Cycloid

History of the Brachistochrone

History of Brach cont

History of Brach cont. 2
(1)  Note:  The crucial steps in this application of the Beltrami Identity are borrowed from the calculus of variations, an upper division undergraduate course.  This derivation applies the Euler-Lagrange formula and is available as  Brachistochrone Part IV.

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Brachistochrone Part  III

Brachistochrone Part