Brachistochronei

Brachistochronei refers to the solutions to the brachistochrone problem. The brachistochrone problem, a fundamental question in calculus of variations, asks for the curve between two points along which a particle, under the influence of gravity and without friction, will travel in the shortest possible time. The solution to this problem is a segment of a cycloid.

A cycloid is the curve traced by a point on the rim of a circle as the circle rolls along a straight line without slipping. The brachistochronei are therefore specific segments of cycloids. The shape of the brachistochronei depends on the starting and ending points.

The problem was first posed by Johann Bernoulli in 1696 and was solved by several mathematicians, including his brother Jacob Bernoulli, Isaac Newton, and Gottfried Wilhelm Leibniz. The discovery of the cycloid as the solution was a significant achievement in the development of calculus of variations. This area of mathematics deals with finding functions that optimize certain integrals, often related to physical phenomena. The brachistochrone problem serves as a classic example and an accessible introduction to these concepts. The mathematical derivation involves minimizing the time integral, which is expressed as an integral of the reciprocal of the particle's speed along the curve.