I'm studying Ergodic Theory and I think I "got" the concept, but I need an example to verify it.
Let's take the simplest possible 2D classical harmonic oscillator whose kinetic energy is $$T=\frac{\dot x^2}{2}+\frac{\dot y^2}{2}$$ and potential energy is $$U=\frac{ x^2}{2}+\frac{y^2}{2}$$
Could you show me the time averages of those two quantities?
Since the amplitudes are unitary we have $$ (x(t),y(t))=(\cos(t+c),\cos(t+d)) $$ for some $c,d\in\mathbb R$ and all $t$. Therefore, $$ \begin{split} \langle U\rangle &=\lim_{\tau\to\infty}\frac1\tau\int_0^\tau\frac12(x(t)^2+y(t)^2)\,dt\\ &=\frac1{2\pi}\int_0^{2\pi}\frac12(x(t)^2+y(t)^2)\,dt\\ &=\frac1{2\pi}\int_0^{2\pi}\frac12(\cos^2(t+c)+\cos^2(t+d))\,dt=\frac12, \end{split} $$ after some simple computations, since $$\int_0^{2\pi}\cos^2(t+c)\,dt=\int_0^{2\pi}\cos^2(t+d)\,dt=\pi. $$ Similarly, $$ \begin{split} \langle T\rangle &=\lim_{\tau\to\infty}\frac1\tau\int_0^\tau\frac12(\dot x(t)^2+\dot y(t)^2)\,dt\\ &=\frac1{2\pi}\int_0^{2\pi}\frac12(\dot x(t)^2+\dot y(t)^2)\,dt=\frac12. \end{split} $$