Let be the origin and some lower point not directly below . Let be the horizontal and vertical distances of the bead from , so that as the bead falls ( as well). The problem stated is to minimize the time that it takes a bead to go from to under gravity alone. The travel time can be expressed as the integral of the inverse of the speed with respect to arclength along the bead's path:
To express the speed in terms of arclength, notice that the potential energy of the bead is where the convention is used, and the kinetic energy is the bead starts from rest, as well. Using conservation of energy you get:
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Rearranging terms gives Changing variables using:
,
where is the increment of length on the curve, the integral for minimization becomes:
where It remains to find the function which minimizes the time.
This problem can be approached using the calculus of variations. In particular, the function is varied until an extreme value for the integral is found. In most cases the Euler-Lagrange equation would be applied next:
However, does not explicitly depend on , and thus the simpler Beltrami identity is used instead:
where is an integration constant. Applying you get:
Squaring both sides yields:
This differential equation is separable and can be integrated to give:
,
where , using and as initial conditions. Using the following definition:
,
the solution can be rewritten parametrically as:
This is a parameterization of a cycloid! To fully solve the problem, and must be determined by satisfying the boundary conditions
,
which in general must be solved numerically.
The actual time can be then expressed in terms of as follows: