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BRACHISTOCHRONE OF GIVEN LENGTH
Problem posed and solved by Jean
Bernoulli in 1718, and studied by Nathan
Moscovitch in 1934.
Other name: isoperimetric brachistochrone. 
The brachistochrone of given length is the curve of length l on which a massive point without initial speed must slide without friction in a uniform gravitational field in such manner that the travel time is minimal among all the curves of length l joining two fixed points O and A (here A(a, b)). 
Initial problem:
minimal, for given .
Differential equation (obtained by applying the EulerLagrange equation): . (see here for a little more detail) Parametrization: where and (for k = 0, we find the usual brachistochrone, in other words the cycloid). Curvilinear abscissa given by: . Travel time given by: . Other parametrization: . 
View of various brachistochrones joining O = (0,
0) and A = (a, 0) for k between –1 and 1, and
a
= 1.
When k approaches –1, the limit curve is the segment line [OA], of length a and infinite travel time. For k = 0, (limit between red and green), we get the cycloid of length and minimal travel time . For k = 1 (bottommost curve), we get a curve of length 5a/2 and travel time (rational curve, as noted by Bernoulli). Relation between c and a: . Length of a complete arch: .


View of various brachistochrones joining O = (0,
0) and A = (a, 0) for k > 1 and a = 1.
Relation between c and a: . Length of a complete arch: .

Remark: these curves also solve the dual problem of determining
the curve of given travel time with minimal length.
Comparison between the curves obtained by solving the differential equation (on the left) and mere scalings of the cycloid (on the right).... 
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© Robert FERRÉOL 2017