# Brachistochrone

Johann Bernoulli posed the following problem in 1696:

If you wanted a ball to roll down a slide1 in the shortest amount of time, what shape should the slide be?

Seeing words like shortest, longest, fastest, etc. in mechanics typically means we need to identify a functional2 and extremize it. This process is an essential component of the calculus of variations, and in the most basic sense, it is similar to finding the minimum/maximum of a function of a single variable in ordinary calculus.

The brachistochrone is the problem of identifying the path with the shortest time of descent.  The time to travel from point $Latex formula$ to point $Latex formula$ is the distance traveled over the velocity. An increment of distance is given by the Pythagorean theorem, such that $Latex formula$, while the velocity can be determined by a conservation of energy. The sum of kinetic and potential energy for a ball3 is $Latex formula$ where $Latex formula$ is its mass, and $Latex formula$ is the acceleration due to gravity. The velocity of the ball is then $Latex formula$. Therefore, the functional we wish to extremize is therefore:

$Latex formula$

To minimize our functional, this term must satisfy the Euler-Lagrange equations

$Latex formula$.

Satisfying the Euler-Lagrange equations (Expand)

Things are a bit easier than they appear at first glance. Since $Latex formula$ is not a function of $Latex formula$, the Euler-Lagrange equations in reduce to the Beltrami identity

$Latex formula$.

Now, let’s evaluate the partial derivative

$Latex formula$

Inserting this into the Beltrami identity results in

$Latex formula$.

This equation can be simplified by multiplying the first term in the parenthesis by one, in the form of $Latex formula$, and factoring. Since the result is equal to a constant, we can make the math easier going forward by taking the reciprocal of this result, squaring it, and calling that a new constant:

$Latex formula$.

By a separation of variables, beginning with $Latex formula$, we can now isolate $Latex formula$.

Which results in the following ordinary differential equation:

$Latex formula$.

Integration via Trigonometric Substitution (Expand)

Since we have successfully separated the variables, the next step is to integrate both sides. However, integrals of roots are often a headache, and in this particular case, a standard change of variables doesn’t help. What we will use is a trigonometric substitution.

Since the square of the numerator added to the square of the denominator is equal to our unknown constant, we can construct a right triangle with $Latex formula$ as the hypotenuse (Fig. 1).

Now, we can write

$Latex formula$.

We can solve for $Latex formula$ by making use of some trigonometric identities, such that

Relevant trigonometric identities (Expand)
$Latex formula$

$Latex formula$.

Since we are looking for the $Latex formula$, we first need $Latex formula$. Differentiating this with respect to $Latex formula$ yields

$Latex formula$.

With this step, we’re almost there, we can now write the differential equation as

$Latex formula$.

Therefore, the differential becomes

$Latex formula$,

which can be readily integrated to give

$Latex formula$.

Since $Latex formula$ must vanish at $Latex formula$, the constant of integration must be $Latex formula$. Finally, if we substitute $Latex formula$, and let $Latex formula$, we arrive the parametric solution.

Integration of both sides does not lead to an equation that can be expressed  as a function $Latex formula$. However, with a substitution for the angle and the unknown constant, the result can be represented by parametric equations.

The solution to the brachistochrone is the parametric equations of a cycloid.

$Latex formula$

The cycloid is the curve traced by a point on a circle rolling in a straight line.

Here is the final result, shown with other curves for comparison:

1. under the action of its own weight, and in the absence of friction
2. a function of functions
3. which begins at rest from the origin