The proof of the vis-viva equation of orbital mechanics found on wikipedia looks, in my opinion, somewhat convoluted and unenlightening. Considering the simplicity and importance of the vis-viva equation, is there a shorter or more insightful derivation?
The vis-viva equation states that for a Kepler orbit of a mass around a central mass $M$, the magnitude of the velocity at any distance $r$ from the center obeys $v^2 = GM(\frac{2}{r}-\frac{1}{a})$, where $a$ is the semi-major axis of the orbit. It is equivalent to saying that the energy of a Kepler orbit with semi-major axis $a$ is $-\frac{GM}{2a}$. This is a very natural generalization of the 'circular' case where the total energy is $-\frac{GM}{2r}$.
Because of the simplicity of the result, I guessed that there could be a relatively straightforward derivation. The derivation on wikipedia involves quite some algebra and, importantly, only holds for elliptical orbits.
$$v^2 = GM\left(\frac{2}{r} - \frac{1}{a} \right)$$
