The Brachistochrone

A classic example of the calculus of variations is to find the brachistochrone, defined as that smooth curve joining two points A and B (not underneath one another) along which a particle will slide from A to B under gravity in the fastest possible time.

$\includegraphics{brachistochrone.svg}$

Using the coordinate system illustrated, we can use conservation of energy to obtain the velocity v of the particle as it makes its descent

$\displaystyle {\frac{{1}}{{2}}}$ mv² = mgx

so that

v = $\displaystyle \sqrt{{2gx}}$ .

Noting also that distance s along the curve s satisfies ds² = dx² + dy², we can express the time T(y) taken for the particle to descend along the curve y = y(x) as a functional:

T(y) = $\displaystyle \int^{B}_{A}$ dt = $\displaystyle \int^{B}_{A}$ $\displaystyle {\frac{{ds}}{{ds/dt}}}$ = $\displaystyle \int^{B}_{A}$ $\displaystyle {\frac{{ds}}{{v}}}$ = $\displaystyle \int^{h}_{0}$ $\displaystyle {\frac{{\sqrt{1+(y')^2}}}{{\sqrt{2gx}}}}$ dx, subject to y(0) = 0, y(h) = a.

The brachistochrone is an extremal of this functional, and so it satisfies the Euler-Lagrange equation

$\displaystyle {\frac{{d}}{{dx}}}$ $\displaystyle \left(\vphantom{ \frac{y'}{\sqrt{2gx(1+(y')^2)}} }\right.$ $\displaystyle {\frac{{y'}}{{\sqrt{2gx(1+(y')^2)}}}}$ $\displaystyle \left.\vphantom{ \frac{y'}{\sqrt{2gx(1+(y')^2)}} }\right)$ = 0, y(0) = 0, y(h) = a.

Integrating this, we get

$\displaystyle {\frac{{y'}}{{\sqrt{2gx(1+(y')^2)}}}}$ = c

where c is a constant, and rearranging

y' = $\displaystyle {\frac{{dy}}{{dx}}}$ = $\displaystyle {\frac{{\sqrt{x}}}{{\sqrt{\alpha-x}}}}$ , with α = $\displaystyle {\frac{{1}}{{2gc^2}}}$ .

We can integrate this equation using the substitution x = αsin²θ to obtain

y = $\displaystyle \int$ $\displaystyle {\frac{{\sqrt{x}}}{{\sqrt{\alpha-x}}}}$ dx = $\displaystyle \int$ $\displaystyle {\frac{{\sin{\theta}}}{{\cos{\theta}}}}$ 2αsinθcosθ dθ = $\displaystyle \int$ α(1 - cos2θ) dθ = $\displaystyle {\frac{{\alpha}}{{2}}}$ (2θ - sin2θ) + k.

Substituting back for x, and using y(0) = 0 to set k = 0, we obtain

y(x) = αsin^-1 $\displaystyle \left(\vphantom{\sqrt{\frac{x}{\alpha}}}\right.$ $\displaystyle \sqrt{{\frac{x}{\alpha}}}$ $\displaystyle \left.\vphantom{\sqrt{\frac{x}{\alpha}}}\right)$ - $\displaystyle \sqrt{{x}}$ $\displaystyle \sqrt{{\alpha-x}}$ .

Definition 5 This curve is called a cycloid.

The constant α is determined implicitly by the remaining boundary condition y(h) = a. The equation of the cycloid is often given in the following parametric form (which can be obtained from the substitution in the integral)

x(θ)	=	$\displaystyle {\frac{{\alpha}}{{2}}}$ (1 - cos2θ)
y(θ)	=	$\displaystyle {\frac{{\alpha}}{{2}}}$ (2θ - sin2θ)

and can be constructed by following the locus of the initial point of contact when a circle of radius α/2 is rolled (an angle 2θ) along a straight line.