Explicit solution for Harmonic Oscillators "solution in quadratures"

Question

So I was reading about integrable systems and of course the harmonic oscillator in one dimension was mentioned as an example. After some simple calculations the author ended up with the following equation $$ t=\int_{q_0}^q\frac{\mathrm{d} q}{\sqrt{2(H(q_0,p_o)-q^2)}} $$ and said that this is solved implicitly by $q$.
My question is: what is the explicit solution? There should exist one, right? I grilled ChatGPT about it and it mentioned elliptic functions but then just lost it. I don't really know much about elliptic functions nor elliptic integrals so any help would be appreciated.

Answer 1

It’s not terribly difficult in this case: the integral is just arcsine, up to some prefactor and integration constant. This recovers $q(t)=A\sin(\omega t+\phi)$ as characteristic for simple harmonic motion (SHM).

The problem becomes significantly harder, though, if we replace a harmonic oscillator $V(q)=q^2$ with some other potential energy function. In particular, the mathematical pendulum effectively replaces $q^2$ with $2(1-\cos q)$. Both versions behave identically for small $q$, so the mathematical pendulum acts like SHM at small angles. But if we want an explicit solution, we’ll have to “solve” the integral obtained by the earlier replacement. This requires us to move to elliptic integrals as a sort of generalized arcsine, and from there to elliptic functions as a generalization of trigonometric functions. So while the mathematical pendulum can be solved analytically, the answer will not look familiar unless you already have background in special functions.