Brachistochrone curve

The curve of fastest descent is not a straight or polygonal line (blue) but a cycloid (red).

In physics and mathematics, a brachistochrone curve (from Ancient Greek βράχιστος χρόνος (brákhistos khrónos) 'shortest time'),[1] or curve of fastest descent, is the one lying on the plane between a point A and a lower point B, where B is not directly below A, on which a bead slides frictionlessly under the influence of a uniform gravitational field to a given end point in the shortest time. The problem was posed by Johann Bernoulli in 1696.

The brachistochrone curve is the same shape as the tautochrone curve; both are cycloids. However, the portion of the cycloid used for each of the two varies. More specifically, the brachistochrone can use up to a complete rotation of the cycloid (at the limit when A and B are at the same level), but always starts at a cusp. In contrast, the tautochrone problem can use only up to the first half rotation, and always ends at the horizontal.[2] The problem can be solved using tools from the calculus of variations[3] and optimal control.[4]

Balls rolling under uniform gravity without friction on a cycloid (black) and straight lines with various gradients. It demonstrates that the ball on the curve always beats the balls travelling in a straight line path to the intersection point of the curve and each straight line.

The curve is independent of both the mass of the test body and the local strength of gravity. Only a parameter is chosen so that the curve fits the starting point A and the ending point B.[5] If the body is given an initial velocity at A, or if friction is taken into account, then the curve that minimizes time differs from the tautochrone curve.

  1. ^ Chisholm, Hugh, ed. (1911). "Brachistochrone" . Encyclopædia Britannica (11th ed.). Cambridge University Press.
  2. ^ Stewart, James. "Section 10.1 - Curves Defined by Parametric Equations." Calculus: Early Transcendentals. 7th ed. Belmont, CA: Thomson Brooks/Cole, 2012. 640. Print.
  3. ^ Weisstein, Eric W. "Brachistochrone Problem". MathWorld.
  4. ^ Ross, I. M. The Brachistochrone Paradigm, in Primer on Pontryagin's Principle in Optimal Control, Collegiate Publishers, 2009. ISBN 978-0-9843571-0-9.
  5. ^ Hand, Louis N., and Janet D. Finch. "Chapter 2: Variational Calculus and Its Application to Mechanics." Analytical Mechanics. Cambridge: Cambridge UP, 1998. 45, 70. Print.