Brachistochrone curve
A Brachistochrone curve is the fastest path for a ball to roll between two points that are at different heights. A ball can roll along the curve faster than a straight line between the points.
The curve will always be the quickest route regardless of how strong gravity is or how heavy the object is. However, it might not be the quickest if there is friction.
The curve can be found using calculus of variations and optimal control.[1]
Brachistochrone Curve Media
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.
Diagrams for Wikipedia entry regarding Galileo's Conjecture
Brachistochrone Bernoulli Direct Method
figure for Brachistochrone curve based on a reading of a proof by Jacob Bernoulli published in Acta Eruditorum in 1697
Bernoulli Challenge to Newton 1
References
- ↑ Ross, I. M. The Brachistochrone Paridgm, in A Primer on Pontryagin's Principle in Optimal Control, Collegiate Publishers, 2009. ISBN 978-0-9843571-0-9.
Other websites
- The straight line, the catenary, the brachistochrone, the circle, and Fermat Unified approach to some geodesics (from 2014)