In February of 1662, Pierre de Fermat wrote a paper Synthesis ad refractiones that explained Descartes-Snell’s Law of light refraction by finding the least time it took for light to travel between two points. This famous approach is now known as Fermat’s principle, and it motivated other searches for minimum principles. A few years earlier, in 1656, Christiaan Huygens had invented the pendulum clock [1], and he began a ten-year study of the physics of the pendulum. He was well aware that the pendulum clock does not keep exact time—as the pendulum swings wider, the period of oscillation slows down. He began to search for a path of the pendular mass that would keep the period the same (and make pendulum clocks more accurate), and he discovered a trajectory along which a mass would arrive at the same position in the same time no matter where it was released on the curve. That such a curve could exist was truly remarkable, and it promised to make highly accurate time pieces.

It made minimization problems a familiar part of physics—they became part of the mindset, leading ultimately to the principle of least action.

This curve is known as a tautochrone (literally: same or equal time in Greek) and Huygens provided a geometric proof in his Horologium Oscillatorium sive de motu pendulorum (1673) that the curve was a cycloid [2]. A cycloid is the curve traced by a point on the rim of a circular wheel as the wheel rolls without slipping along a straight line. Huygens invented such a pendulum in which the mass executed a cycloid curve. It was a mass on a flexible yet inelastic string that partially wrapped itself around a solid bumper on each half swing. In principle, whether the pendulum swung gently, or through large displacements, the time would be the same. Unfortunately, friction along the contact of the string with the bumper prevented the pendulum from achieving this goal, and the tautochronic pendulum did not catch on.

Fig. 1 Huygens’ isochronous pendulum.  The time it takes the pendulum bob to follow the cycloid arc is independent of the pendulum’s amplitude, unlike for the circular arc, as the pendulum slows down for larger excursions.

The solution of the tautochrone curve of equal time led naturally to a search for the curve of least time, known as the brachistochrone curve for a particle subject to gravity, like a bead sliding on a frictionless wire between two points. Johann Bernoulli published a challenge to find the brachistochrone in 1696 in the scientific journal Acta Eruditorum that had been founded in 1682 by Leibniz in Germany in collaboration with Otto Mencke. Leibniz envisioned the journal to be a place where new ideas in the natural sciences and mathematics could be published and disseminated rapidly, and it included letters and commentaries, acting as a communication hub to help establish a community of scholars across Europe. In reality, it was the continental response to the Proceedings of the Royal Society in England.  Naturally, the Acta and the Proceedings would later take partisan sides in the priority dispute between Leibniz and Newton for the development of the calculus.

When Bernoulli published his brachistochrone challenge in the June issue of 1696, it was read immediately by the leading mathematicians of the day, many of whom took up the challenge and replied. The problem was solved and published in the May 1697 issue of the Acta by no less than five correspondents, including Johann Bernoulli, Jakob Bernoulli (Johann’s brother), Isaac Newton, Gottfried Leibniz and Ehrenfried Walther von Tschirnhaus. Each of them varied in their approaches, but all found the same solution. Johann and Jakob each considered the problem as the path of a light beam in a medium whose speed varied with depth. Just as in the tautochrone, the solved curve was a cycloid. The path of fastest time always started with a vertical path that allowed the fastest acceleration, and the point of greatest depth always was at the point of greatest horizontal speed.

The brachistrochrone problem led to the invention of the variational calculus, with first steps by Jakob Bernoulli and later more rigorous approaches by Euler.  However, its real importance is that it made minimization problems a familiar part of physics—they became part of the mindset, leading ultimately to the principle of least action.

Mathematical Description of the Tautochrone Curve

If the length of the pendulum is L, the position coordinates (relative to the cusp of the cycloid) are

where the angle θ is between the straight section of the string and the vertical and depends on time as

for a maximum angle θ₀. The angular frequency of the isochronous pendulum is given by the standard expression for a small-amplitude pendulum as

For small-angle oscillations, the tautochrone and the conventional pendulum have the same periods and amplitudes. The difference is that the tautochrone has that same period regardless of the amplitude, so no approximation is needed.

Huygens’ proof of the tautochrone curve was made geometrically without the use of calculus, requiring almost 18 pages and 16 figures.

The modern proof (due to Euler) takes only a few lines of calculus. All that is needed is to recognize that the equation of motion of the pendulum bob along its path of length s should be that of a simple-harmonic oscillator

because a simple harmonic oscillator always oscillates at the same frequency regardless of amplitude. Of course, a simple pendulum is not a simple harmonic oscillator because it becomes nonlinear and slows down for large amplitudes. The right-hand-side is the standard force on the bob given by the gradient of the potential energy along the curve

yielding the simple differential equation

which is integrated to give a quadratic dependence of height on path length

To get an equation for the curve itself, plug this back in to the differential equation and square it to give

Euler recognized this as the differential equation of a cycloid

Notes

[1] Galileo conceived the pendulum clock in 1641, and his son Vincenzo started construction, but it was never finished.  Huygens submitted and received a patent in 1657 for a practical escape mechanism on pendulum clocks that is still used today.

[2] Huygens, Christiaan; Blackwell,, Richard J., trans. (1986). Horologium Oscillatorium (The Pendulum Clock, or Geometrical demonstrations concerning the motion of pendula as applied to clocks). Ames, Iowa: Iowa State University Press