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Brachistochrone curve
In mathematics, a brachistochrone curve (), or curve of fastest descent, is the curve that would carry an idealized point-like body, starting at rest and moving along the curve, without friction, under constant gravity, to a given end point in the shortest time. For a given starting point, the brachistochrone curve is the same as the tautochrone curve.
==The brachistochrone is the cycloid==
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.
The problem can be solved with the tools from the calculus of variations and optimal control.〔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.〕
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.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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