The classical (undamped) pendulum ODE is
$$\ddot \theta = -\frac{g}{\ell} \sin(\theta)$$
Defining state ${\bf x} := (\theta, \dot \theta)$, we have a system of $2$ ODEs, $\dot {\bf x} = {\bf f} ({\bf x})$, where ${\bf f} : \Bbb R^2 \to \Bbb R^2$.
Consider the change of coordinates ${\bf y} = {\bf \Phi} ({\bf x})$, where vector field ${\bf \Phi} : \Bbb R^2 \to \Bbb R^2$ is a diffeomorphism. Differentiating with respect to time,
$$\dot {\bf y} = \left(\left({\bf D \, \Phi} \cdot {\bf f} \right) \circ {\bf \Phi}^{-1} \right) ({\bf y})$$
where ${\bf D \, \Phi}$ denotes the Jacobian of vector field ${\bf \Phi}$, $\cdot$ denotes multiplication, $\circ$ denotes composition and ${\bf \Phi}^{-1}$ denotes the inverse of ${\bf \Phi}$. Are there "nice" choices of vector field ${\bf \Phi}$ that make vector field $$\color{blue}{\left( {\bf D \, \Phi} \cdot {\bf f} \right) \circ {\bf \Phi}^{-1}}$$ "nice" in some sense?
What is "nice"? For example, polynomial and rational functions are nice. All elementary functions are nice. Anything of relatively "low" descriptive complexity would be nice. In fact, any alternative to the classical pendulum ODE would be nice, even if highly contrived.
My work
The only idea I had was to rewrite the pendulum ODE in Cartesian coordinates, which led to
$$\begin{aligned} (\ddot x – g) y – x \ddot y &= 0\\ x^2 + y^2 &= \ell^2 \end{aligned}$$
which isn't really a system of ODEs. Rather, it is a system of differential-algebraic equations (DAEs). Nothing else occurred to me. I am not familiarized with diffeomorphisms. Ideas are most welcome.
Best Answer
It is possible to find another way for the solution of $$y''+k\sin(y)=0 \qquad \text{with} \qquad y=y(x)\qquad \text{and} \qquad k>0$$ Switch variables and then write $$\frac {x''}{[x']^3}=k\sin(y)$$ The usual reduction of order gives $$x'=\pm\frac{1}{\sqrt{2 k \cos (y)+c_1}}$$ So $$x+c_2=\pm\frac{2}{\sqrt{c_1+2 k}}F\left(\frac{y}{2}|\frac{4 k}{c_1+2 k}\right)$$ where appears the elliptic integral of the first kind.
This can be inversed easily leading to the amplitude for the Jacobi elliptic functions.