Nonlinearity Affects a Pendulum

The equation of motion for a pendulum is the differential equation

$$\frac{d^2\theta}{dt^2} + \frac{g}{\ell} \sin \theta = 0$$

where g is the acceleration due to gravity and ℓ is the length of the pendulum. When this is presented in an introductory physics class, the instructor will immediately say something like “we’re only interested in the case where θ is small, so we can rewrite the equation as

$$\frac{d^2\theta}{dt^2} + \frac{g}{\ell} \theta = 0$$

Questions

This raises a lot of questions, or at least it should.

• Why not leave sin θ alone?
• What justifies replacing sin θ with just θ?
• How small does θ have to be for this to be OK?
• How do the solutions to the exact and approximate equations differ?

First, sine is a nonlinear function, making the differential equation nonlinear. The nonlinear pendulum equation cannot be solved using mathematics that students in an introductory physics class have seen. There is a closed-form solution, but only if you extend “closed-form” to mean more than the elementary functions a student would see in a calculus class.

Second, the approximation is justified because sin θ ≈ θ when θ is small. That’s true, but it’s kinda subtle.

The third question doesn’t have a simple answer, though simple answers are often given. An instructor could make up an answer on the spot and say “less than 10 degrees” or something like that. A more thorough answer requires answering the fourth question.

Longer period

The primary difference between the nonlinear and linear pendulum equations is that the solutions to the nonlinear equation have longer periods. The solution to the linear equation is a cosine. Solving the equation determines the frequency, amplitude, and phase shift of the cosine, but qualitatively it’s just a cosine. The solution to the corresponding nonlinear equation, with sin θ rather than θ, is not exactly a cosine, but it looks a lot like a cosine, only the period is a little longer.

OK, the nonlinear pendulum has a longer period, but how much longer? The period is increased by a factor f(θ0) where θ0 is the initial displacement. The exact answer depends on a special function called the “complete elliptic integral of the first kind.”

Linear solution with adjusted period

Since the nonlinear pendulum equation is roughly the same as the linear equation with a longer period, you can approximate the solution to the nonlinear equation by solving the linear equation but increasing the period. How good is that approximation?

Let’s do an example with θ0 = 60° = π/3 radians. Then sin θ0 = 0.866 but θ0, in radians, is 1.047, so we definitely can’t say sin θ0 is approximately θ0. Assume the pendulum starts from rest, i.e. θ'(0) = 0.

Obviously the solution to the nonlinear equation has a longer period. In fact it’s 7.32% longer. When comparing the solution of the nonlinear equation and the solution to the linear equations with the period stretched by 7.32%, the solutions differ by very little.

Update: Part of the observed difference in numerical models can be attributed to numerical error from solving the pendulum equation. When using exact solutions like Jacobi functions, the error is even smaller and periodic, as expected. This suggests that the nonlinearity primarily affects the period rather than the fundamental shape of the oscillation.