A brachistochrone curve (Gr. βράχιστος, brachistos - the shortest, χρόνος, chronos - time) or curve of fastest descent, is the path that will carry a point-like body from one place to another in the least amount of time. The body is released at rest from the starting point and is constrained to move without friction along the curve to the end point, while under the action of constant gravity. The brachistochrone curve is the same as the tautochrone curve for a given starting point.
Given two points A and B, with A not lower than B, only one upside down cycloid passes through both points, has a vertical tangent line at A, and has no maximum points between A and B: the brachistochrone curve. The curve does not depend on the body’s mass or on the strength of the gravitational constant.
If the body is given an initial velocity at A, or if friction is taken into account, then the curve that minimizes time will differ from the one described above.
According to Fermat’s principle: The actual path between two points taken by a beam of light is the one which is traversed in the least time. In 1697 Johann Bernoulli used this principle to derive the brachistochrone curve by considering the trajectory of a beam of light in a medium where the speed of light increases following a constant vertical acceleration (that of gravity g).
The Conservation of energy can be used to express the speed of a body in a constant gravitational field as:
where y represents the vertical distance the body has fallen. The speed of motion of the body along an arbitrary curve does not depend on the horizontal displacement.
where vm is the constant and represents the angle of the trajectory with respect to the vertical.
The equations above allow us to draw two conclusions:
Simplifyingly assuming that the particle (or the beam) with coordinates (x,y) departs from the point (0,0) and reaches maximum speed after a falling a vertical distance D:
Rearranging terms in the law of refraction and squaring gives:
which can be solved for dx in terms of dy:
Substituting from the expressions for v and vm above gives:
Johann’s brother Jakob showed how 2nd differentials can be used to obtain the condition for least time. A modernized version of the proof is as follows. If we make a negligible deviation from the path of least time, then, for the differential triangle formed by the displacement along the path and the horizontal and vertical displacements,
On differentiation with dy fixed we get,
And finally rearranging terms gives,
where the last part is the displacement for given change in time for 2nd differentials. Now consider the changes along the two neighboring paths in the figure below for which the horizontal separation between paths along the central line is d2x (the same for both the upper and lower differential triangles). Along the old and new paths, the parts that differ are,
For the path of least times these times are equal so for their difference we get,
And the condition for least time is,
Babb, Jeff; Currie, James (July 2008), "The Brachistochrone Problem: Mathematics for a Broad Audience via a Large Context Problem", TMME 5 (2&3): 169–184
D.T.Whiteside, Newton the mathematician, in Bechler, Contemporary Newtonian Research, p. 122.
References & Further Reading