Calculus of variations and optimal control; optimization Classical problems Problem

Brachistochrone problem

Builds on Integral functional with a Lagrangian density Beltrami identity Euler–Lagrange equation

Problem

Let $A$ and $B$ be two points in a vertical plane, with $B$ lower than $A$ and not on the vertical line through $A$ . A particle starts at rest at $A$ and slides without friction along a curve from $A$ to $B$ under constant gravity $g$ . Which curve makes the time of travel from $A$ to $B$ as small as possible?

In coordinates, put $A = (0, 0)$ , measure $y$ downwards, and let $B = (x_1, y_1)$ with $x_1, y_1 > 0$ . Energy conservation gives the speed $\sqrt{2 g y}$ . So for a graph $y(x)$ the time is the integral functional $T(y) = \int_0^{x_1} \sqrt{\frac{1 + y'(x)^2}{2 g \, y(x)}} \, dx,$ to be minimised among curves with $y(0) = 0$ and $y(x_1) = y_1$ .

Remarks

The answer is an arc of a cycloid, the curve traced by a point on a circle that rolls along a horizontal line. Its parametric form is $x = r(\theta - \sin \theta)$ , $y = r(1 - \cos \theta)$ , with the radius $r$ fixed by the condition that the curve passes through $B$ . The integrand does not depend on $x$ , so the Beltrami identity applies and gives $y (1 + y'^2) = 2r$ for a constant $r$ ; the cycloid solves this equation. The fastest curve is not the straight segment; when $x_1 / y_1 > \pi / 2$ , the cycloid even dips below the height of $B$ before it rises to $B$ . Johann Bernoulli posed the problem as a public challenge in Acta Eruditorum; solutions by Johann Bernoulli, Jacob Bernoulli, Leibniz and Newton appeared in 1697. The problem is often taken as the start of the calculus of variations.

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Euclidean space object
Euler–Lagrange equation concept
Function concept
Integral functional with a Lagrangian density concept
Function of several variables concept
Partial derivative concept
Beltrami identity statement
Brachistochrone problem problem

Tag	`ZWQ5JBR6XF`
URI	`m2://ZWQ5JBR6XF`
Kind	problem
Writing	W0 stub
MSC	49K05, 49-03, 70H03
Status	This copy of the site has no build data. Run `m2 status ZWQ5JBR6XF`.
About	Calculus of variations
Uses	Integral functional with a Lagrangian density, Beltrami identity, Euler–Lagrange equation

Brachistochrone problem

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