Brachistochrone

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 $A$ and $B$ and lying in a common vertical plane ($B$ below $A$) to find the one along which a material point moving from $A$ to $B$ solely under the influence of gravity, would arrive at $B$ within the shortest possible time. The problem can be reduced to finding a function $y(x)$ that constitutes a minimum of the functional

$$J(y)=\int\limits_a^b\sqrt{\frac{1+y'^2}{2gy}}dx,$$

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

References

[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)

Comments

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].

References

[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)