Good afternoon, I have some trouble proving periodicity with a conservative system. The problems is that I don't know how to make a formal demonstration although I can see it quite true.
I have the following system: $$\frac{dx}{dt}= y \\ \frac{dy}{dt}= -g(x)$$
We know that g is a strictly increasing function.
And $g(0)=0$ , $\lim_{x \to \infty} \int^{x}_{0}g(u)du = \infty$
Doing some algebra it is clear that the phase path are given by: $$y=\pm[2(C-V(x))]^{1/2}$$
where $V(x)=\int^{x}_{0} g(u)du$
With the help of the information of $g$ we can see that the phase path must look like ovals. So the orbits are closed paths, and so we can say that orbits are periodic.
But how can I prove this ? I was thinking in giving a diffeomorphism from the path to the circle but I cant find it.
Thanks for you help and your time.