Chapter 1: The Challenge of the Brachistochrone

In June of 1696, the prominent mathematician Johann Bernoulli presented a fascinating problem in the Acta Eruditorum, the first German scientific journal. The problem is stated as follows: "Given two points A and B in a vertical plane, determine the path AMB that a body M, descending solely under the influence of gravity from point A, would take to reach point B in the shortest duration." A simpler interpretation would be: "What is the curve traced by a particle affected only by gravity that starts at point A and arrives at point B in the least time?"

The image below depicts Johann Bernoulli (left) alongside the original statement of the problem from the June 1696 issue of Acta Eruditorum (right).

This intriguing mathematical quandary is referred to as the brachistochrone problem. Although Bernoulli had already devised a solution, he invited mathematicians across Europe to tackle it, granting them a six-month period for their responses. Unfortunately, no solutions emerged by the deadline, not even from the esteemed Gottfried Leibniz, who asked for additional time.

On January 29, 1697, Isaac Newton received Bernoulli's challenge in the mail and solved it overnight, sending his answer anonymously. Upon receiving the solution, Bernoulli famously proclaimed that he recognized the genius behind it "tanquam ex ungue leonem," or "as the lion by its claw."

The translation of Newton's handwritten solution reads as follows:

"From point A, draw an unbounded straight line APCZ parallel to the horizontal. Upon this line, describe any cycloid AQP that intersects the straight line AB at point Q. Then, create another cycloid ADC such that the ratio of its base and height [as AC : AP] corresponds to the ratio of the previous cycloid's base and height as AB to AQ. This new cycloid will pass through point B and represents the path along which a heavy body will descend most rapidly from A to B."

For a detailed exploration of Newton's solution requested by the Scottish mathematician David Gregory, refer to this link.

Chapter 2: Understanding the Brachistochrone Curve

The brachistochrone, often referred to as the curve of fastest descent, is a specific path in a two-dimensional plane where a bead moves from an initial point A to a lower point B, sliding down in the shortest possible time due to gravitational forces.

The following assumptions apply to this problem:

• Friction is negligible.
• The bead begins from a state of rest.
• The gravitational field remains constant (g).

Modern mathematical approaches define the solution as a function y = y(x), with point A positioned at (0,0) and point B located at (a, b). Starting from rest, we can apply energy conservation principles:

Equation 1: Energy conservation.

We can express dt as follows:

Equation 2: Infinitesimal interval dt in relation to x and y.

The total time taken for the bead to travel from A = (0,0) to B = (a,b) is given by:

Equation 3: Total time interval T for the bead's journey.

This time T is dependent on the function y(x) and is classified as a functional (a function of a function). Unlike standard functions that depend on variables, a functional relies on entire functions.

Our objective is to determine which function y(x) minimizes the total time T, necessitating an understanding of a mathematical field known as calculus of variations.

Chapter 3: The Calculus of Variations

Let us consider a function ψ(x) satisfying the conditions illustrated below, where ψ(x₀) = y₀ and ψ(x₁) = y₁. Introduce a second function, closely related to the first, expressed as:

Equation 4: A nearby function u(x) with specific conditions.

To satisfy the conditions on ψ(x), the constraints on u(x) must hold true.

Next, we can examine the following functional:

Equation 5: A functional with integrand L influenced by x, y, and y'.

By altering L(x, y, y'), we derive varying outcomes for the definite integral S[y(x)]. Now, consider the variation of L as we adjust ψ(x):

Equation 6: Variation of L with respect to ψ(x) and ψ'(x).

Integrating both sides, performing integration by parts on the second term, and utilizing the conditions on u(x), we arrive at the following expression for the integral's variation:

Equation 7: Variation of the integral S post-adjustment.

For omitted steps prior to Equation 7, refer to Butkov. If S is minimal, then δS = 0 (as in standard calculus). Since u(x) is arbitrary, we must have:

Equation 8: Condition for δS = 0.

The expression within the brackets vanishes when y(x) aligns with the function ψ(x), allowing L to reach an extremum. This leads us to the renowned Euler-Lagrange equation:

Equation 9: The Euler-Lagrange equation.

Applying this to find the minimum time in Equation 3 yields:

Equation 10: The integrand of Equation 3 to substitute into the Euler-Lagrange equation.

(Note that constants were omitted as they will cancel out). After a series of algebraic manipulations, we obtain the following differential equation and its corresponding solution:

Equation 11: The parametric equations defining a cycloid.

Here, k is a constant dependent on boundary conditions, and we utilized the following variable transformation:

Equation 12: Variable change employed to derive Equation 11.

These parametric equations depict a cycloid, representing the curve that minimizes T, as illustrated below.

Chapter 4: Newton's Further Contributions

In 1699, Nicolas Fatio de Duillier, a mathematician and natural philosopher, released a treatise titled "Double geometric research on the curve of the fastest descent," which provided another solution to the brachistochrone problem.

Once again, David Gregory sought Newton's assistance in simplifying Fatio's solution. The condensed version, consisting of only twelve lines (excluding the crossed-out incorrect lines), was forwarded by Newton to Gregory in 1700. This section aims to elucidate this simplified solution, closely aligned with Roy's analysis.

In the diagram below, we clarify the essential quantities involved. Following Newton’s approach, we start with:

From fundamental kinematics, the velocity of the falling bead at point x is determined by:

Equation 13: Velocity of the bead as a function of height x.

The time required for the bead to traverse from E to G is proportional to:

Equation 14: Time for the bead's motion from E to G.

Defining:

Equation 15: Definitions of R² and S².

Minimizing the overall time t concerning q leads us, after straightforward algebra, to the cycloid's differential equation, represented by Equation 11.

Thank you for reading! As always, I appreciate constructive feedback and suggestions.

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