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
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.
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.