Closed-form solution to the nonlinear pendulum equation

The previous post looks at the nonlinear pendulum equation and what difference it makes to the solutions if you linearize the equation. If the initial displacement is small enough, you can simply replace sin θ with θ. If the initial displacement is larger, you can improve the accuracy quite a bit by solving the linearized equation and then adjusting the period. You can also find an exact solution, but not in terms of elementary functions; you have to use Jacobi elliptic functions. These are functions somewhat analogous to trig functions, though it’s not helpful to try to pin down the analogies. For example, the Jacobi function sn is like the sine function in some ways but very different in others, depending on the range of arguments. We start with the differential equation θ″(t) + c² sin( θ(t) ) = 0 where c² = g/L, i.e. the gravitational constant divided by pendulum length, and initial conditions θ(0) = θ0 and θ′(0) = 0. We assume −π < θ0 < π. Then the solution is θ(t) = 2 arcsin( a…