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).

Fig. 1
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.

Created by Zorgit , CC BY-SA 3.0
Created by Zorgit , License: CC BY-SA 3.0

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

Created by Larry Phillips.
Created by Larry Phillips.

  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