Geometrical Beauty: The Properties of the Tautochrone Curve
By: Marco Tavora Ph.D.
Tags: Math

The discovery of the form of the tautochrone, “the curve for which the time taken by an object sliding (without friction) in a uniform gravitational field to its lowest point is independent of its starting point,” was made in 1659 by the Dutch physicist, mathematician, astronomer, and inventor Christiaan Huygens. Huygens is widely regarded as one of the greatest scientists in history.

Figure 1: Huygens and the front page of his book Horologium Oscillatorium (source).

The Cycloid

His geometrical proof that the curve described above is an inverted cycloid (see the definition below) was published in his book Horologium Oscillatorium: Sive de Motu Pendulorum ad Horologia Aptato Demonstrationes Geometricae (“The Pendulum Clock: or geometrical demonstrations concerning the motion of pendula as applied to clocks”) published in 1673. The Horologium Oscillatorium is considered one of the three greatest works on mechanics in the 17th century, together with Galileo Galilei’s Discourses and Mathematical Demonstrations Relating to Two New Sciences (1638) and Isaac Newton’s Philosophiæ Naturalis Principia Mathematica (1687).

The cycloid is by definition, the curve traced by a point on a circle as it rolls along a line. It is the red curve shown in Figure 2.

Figure 2: The cycloid is the curve traced by a point on the circle as it rolls along a line (source).

Interestingly, this property of the cycloid was mentioned in Herman Melville’s book “Moby Dick” (1851) in the following passage: “[The try-pot] is also a place for profound mathematical meditation. It was in the left-hand try-pot of the Pequod, with the soapstone diligently circling round me, that I was first indirectly struck by the remarkable fact, that in geometry all bodies gliding along a cycloid, my soapstone, for example, will descend from any point in precisely the same time.”

Figure 3: The cycloid is mentioned by the American writer Herman Melville in his 1851 book “Moby-Dick; or, The Whale.”(source).

Calculating the Time

We will now use basic calculus techniques to prove that the tautochrone curve is an inverted catenary. After Huygens’s proof, the conjecture was demonstrated by many other prominent mathematicians (including, among others, Lagrange and Niels Henrik Abel) using different techniques. This section will be based on Simmons.

Figure 4: Newton’s cradle demonstrates conservation of energy (and momentum) using swinging spheres (source).

Suppose the bead is released from rest. The law of conservation of energy gives us (the variable y is shown in see Fig. 5):

Equation 1: Total energy Conservation.

K and T are respectively the kinetic and potential energies of the bead (at the origin O and then at some non-specified y). As will be shown shortly, the beads released at O, A, and C (Fig.5) will arrive at B simultaneously.

Figure 5: The beads released in O, A, and C will arrive at B simultaneously (based on Simmons).

Using the infinitesimal version of Pythagoras theorem,

Equation 2: Line element.

and isolating the time variable, the right-hand side of Eq. 1 becomes:

Equation 3: Time it takes for the bead to go from O to the point (x, y).

where τ was chosen to represent the time the bead takes to go from O to (x,y).

Substituting Eq. 4 below (the parametric representation of a cycloid)

Equation 4: Parametric equation for the catenary.

in Eq. 3 and then canceling the (1-cos θ) factor, a trivial integration gives us:

Equation 5: The time for the bead to slide down the cycloid by an arc corresponding to the angle θ.

Now, we need to prove that τ is independent of θ. In order to do that we use the following equation from basic kinematics:

Equation 6: The dependence of ds/dt on the vertical coordinates y and y₀ (often referred to as Torricelli equation). The constant y₀ is the initial rest position of the bead.

and change θ for (θ-θ₀). The interval τ becomes:

Now making the substitution

we obtain the integral of the inverse of the sine function which is trivial.

Figure 6: Four balls sliding a cycloid from starting at different positions, arrive at the bottom at the same time. (on the top is the time-space diagram).

Our final result is then:

Equation 7: The time it takes for the bead to slide down the cycloid does not depend on its initial position.

We, therefore confirm, as expected, that the time it takes for the bead to slide down a cycloidal wire is constant and does not depend on its initial position (see Fig. 6).