Time averages for a 2-dimensional harmonic oscillator

[Lo Scrondo]

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 $$U=\frac{ x^2}{2}+\frac{y^2}{2}$$.

I'd like to find the time averages of the two quantities. My intuition is that they arent't qualitatively different from the one-dimensional case, but I'd really welcome some help

 

[vanhees71]

You'll find that the oscillator has harmonic solutions and then the time average is
$$\langle U \rangle=\frac{1}{T} \int_0^T \mathrm{d} t \frac{1}{2} (x^2+y^2),$$
where $T$ is the period of the harmonic motion, and analogously for the kinetic energy.

So just write down the general solution of your equations of motion and calculate the integrals. It's not too difficult.