The brachistochrone problem, posed by Johann Bernoulli in 1696, asks: which path allows a bead to slide fastest between two points under gravity? Surprisingly, the answer is not the straight line, but a special curved path called a cycloid. Let's watch two beads race - one on the straight path, one on the curved path.
To solve the brachistochrone problem, we need key physics principles. Conservation of energy tells us that potential energy converts to kinetic energy as the bead falls. This gives us the velocity formula: v equals square root of 2gy, where y is the vertical distance fallen. Notice how the bead accelerates as it drops - gaining speed with greater vertical descent.
Now we formulate the problem mathematically. Total time equals the integral of distance over velocity. The arc length element ds equals square root of 1 plus dy dx squared times dx. Combined with our velocity formula, we get the time integral that must be minimized. This is a calculus of variations problem - finding the function that minimizes a functional.
We apply the Euler-Lagrange equation to our functional. This powerful tool from calculus of variations gives us the differential equation that the optimal curve must satisfy. After working through the partial derivatives and simplifications, we obtain a first integral - a conserved quantity that characterizes the brachistochrone curve.
The solution is a cycloid - a curve traced by a point on a circle rolling along a straight line. Its parametric equations involve the angle theta and radius a. Watch as the rolling circle generates this elegant curve. The cycloid's steep initial descent provides rapid acceleration, while its gentler slope maintains high speed - making it the fastest path between two points under gravity.