the point-mass. Its kinetic energy equals mv(y)2/2,

from conservation of energy that

(1.9) v(y) = v^2/ .

Thus the speed of the point-mass depends only on its

dinate.

Consider the medium described by equation (1.9).

the Fermat principle, the desired curve is the trajectory

A to B. One can approximate the continuous medium

one consisting of thin horizontal strips in which the sp

constant. Let f i , ^ , - •• be the speeds of light in the

etc., strips, and let ^i,a2,... be the angles made by

of light (a polygonal line) with the horizontal border

consecutive strips. By Snell's law, cos ai/vi = cos

figure 1.10. Thus, for all i,

(1.10) ^ ^ = const.

Now return to the continuous case. Taking (1.9) into

tion (1.10) yields, in the continuous limit:

/ H H .N cosa(v)

(1.11) ^ = const.

Vv

Taking into account that tan a = dy/dx, equation

differential equation for the brachistochrone y' — \/(C

equation can be solved, and Johann Bernoulli knew t

solution is the cycloid, the trajectory of a point on a ci

without sliding, along a horizontal line; see figure 1.13

In fact, the argument proving equation (1.11) giv

One does not have to assume that the speed of light

only. Assume, more generally, that the speed of light at

given by a function v(x,y) (so it does not depend on the

the medium is anisotropic). Consider the level curves

v and let 7 be a trajectory of light in this medium. Let

of light along 7 considered as a function on this curv

Incidentally, the cycloid also solves another problem: to find

that a mass point, sliding down the curve, arrives at the end point B

no matter where on the curve it started.