The Wolfram Functions page

Jacobi elliptic function dn: Theorems gives an

exact analytic solution for the position

and

of a pendulum, assuming that the clock starts when the pendulum is at its lowest point. Here

is the angular amplitude,

is the length of the simple pendulum, and

is the acceleration due to gravity.

,

.

To make the pendulum animation loop properly, we need to know how the period of a pendulum depends on its amplitude, so we can choose proper amplitudes. Now, a small-angle pendulum is approximately a harmonic oscillator, and its period is

, which does not depend on the angular amplitude. But for a large-angle pendulum, we have to figure out just how the period does depend on the amplitude. If we knew the angular velocity

as a function of the angle

, then we could integrate

1/ω for one complete oscillation. Or even better, integrate it for 1/4 of an oscillation, that is, from

down to 0; then we do not have to worry about the changing signs for

and

. We can figure out

by using conservation of mechanical energy. (Actually, we use mechanical energy per unit mass to remove the mass from consideration.)

The potential energy is

, so the potential energy per unit mass is

. For convenience, we measure the height from the pivot point, not the lowest point (it does not matter). This means that the height

, so

.

The total mechanical energy

is constant in this system. The pendulum has zero kinetic energy when it is at its highest point, that is, when

θ=α. So

only depends on the amplitude. Working per unit mass,

.

The kinetic energy is then

, so

.

The formula for kinetic energy is

, so that

, and

. Notice that we only get the magnitude here.

And

. Again, this is only the magnitude.

.

Now we are ready to integrate. We look at only 1/4 of a complete oscillation, starting from rest when

θ=α, and ending when

θ=0. During this part of the motion,

is negative because

is decreasing.

.

We would particularly like to know the ratio of the period to the period of the small-angle pendulum:

.

.

The amplitudes in the Demonstration are chosen so that their

's are in the ratios (1, 5/4, 5/3, 5/2); this means that they will complete 5, 4, 3, and 2 oscillations in one complete cycle.