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BRACHISTOCHRONE CURVE
Curve studied and named by Jean Bernoulli in 1718 and
Euler in 1736.
From the Greek brakhisto "the shortest" (therefore written with an "i" and not a "y") and chronos "time". See also wwwgroups.dcs.stand.ac.uk/~history/HistTopics/Brachistochrone.html 
Functional equation:
minimal (expression coming from the fact that the
speed of the moving body is proportional to ).
Differential equation (obtained by
applying the EulerLagrange
equation): .

The brachistochrone (curve) is the curve on which a massive point without initial speed must slide without friction in an uniform gravitational field in such manner that the travel time is minimal among all the curves joining two fixed points O and A (here A(a,b)).
Solution for a > 0 (result found at the same time by Leibniz, Newton, L'Hospital, Jean and Jacques Bernoulli): an arc of a cycloid starting with a vertical tangent.
It can be noted that if the slope b/a between O and A is less than 2/p »63%, corresponding to an angle of »32 ° with the horizontal, the fastest curve has, as in the figure above, a portion that goes upwards!
This is valid in the limit when the points O and A are at the same altitude, in which case the straight line would have an infinite travel time.
For , the travel time is:  for the cycloid,  for an arc of a circle (not represented opposite),  for the straight line,  for a free fall followed by a horizontal path (beaten by the straight line, contrary to the situation of the animation at the top of the page). 
Here the two balls are side by side for a while, then
the blue one follows a straight horizontal line. The cycloid is still the
fastest!
Animation dedicated to editorial board of the TV show "incredible experiments" on France 2 that struggles to accept this phenomenon... 

The way Jean Bernoulli posed the problem was slightly different: find the curve minimising the travel time from a point O, without initial speed, to a vertical plane (at an indefinite point). The solution is half an arch of a cycloid with a vertical start and horizontal end, perpendicular to the plane. At the end point, the moving body's altitude decreased by times the distance of O to the plane. 
the red cycloid beats the other two 
It can also be asked what the brachistochrone curve among
the curves joining two points and having a given shape would be.
For example, for two points at the same altitude and Vshaped curves, the brachistochrone curve is the one for which the angle of the V is a right angle, as is shown in the animation opposite. This example also shows that among all the linear paths starting at a given point A and ending on a given vertical plane P, the brachistochrone is the one making a 45° angle with the horizontal and that is traced in the vertical plane passing through A and orthogonal to P (result obtained by Galileo in 1638). The problem of the brachistochrone with given length is studied on this page. 

One can also try to find the brachistochrone "with friction".
We get the parametrization
where
is the coefficient of friction. Opposite, the brachistochrone without friction
(hence the cycloid) is in red.
See this article. 
Do the creators of skateboard ramps know that the fastest ramp has the shape of a cycloid? The answer is yes according to the article from which the opposite pictures are taken, but apparently, no cycloidal ramp was ever created. 

We can also look for the brachistochrone curves obtained
for various
speed laws.
Here ,
but for the more general case ,
the differential equation of the brachistochrone is .
If ,
the brachistochrone we get this time is an arc of a circle.
If now the speed only depends on the distance to O
(), the differential
equation of the brachistochrone is .
For ,
the brachistochrone is the logarithmic
spiral ; this case corresponds to a moving body in a reference frame
in uniform rotation around O (hence subject to the centrifugal force)
when the speed at O is zero.
If there is zero speed at distance a to O,
then
and the brachistochrone is an epicycloid.
This can be applied to the shape of the chistera for Basque pelota in order
for the ball to have maximum speed when leaving the glove.
The case is that of a moving body subject to Earth's gravitation (a being the radius of the Earth) and is solved by an hypocycloid ; therefore, it is the shape of a tunnel dug underground that would minimise the time of travel between one point and another of the surface, when only gravity is taken into account. See for example mathworld and this text. 
In red, the brachistochrone tunnel. 
For other curves describing the movement of a massive
point in a gravitational field under certain conditions, see isochrone
curve of Huygens, tautochrone
curve,
synchronous curve,
synodal
curve, et L'Hospital
quintic.
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© Robert FERRÉOL 2017