# Two Independent Harmonic Oscillators is NOT Ergodic!

I read on a book that the system of two or more independent harmonic oscillators in classical mechanics is not ergodic. I want to know why a harmonic oscillator is actually ergodic but two or more ones is not. Is it related to this fact that the phase space of two independent harmonic oscillators is the product $M \times M$ and each one's ergodicity does not force the whole ergodicity?

• Could you put the name of the book as well? – AHusain Mar 6 '18 at 6:38
• @AHusain Ergodic Features of Harmonic Oscillator Systems – mathvc_ Mar 6 '18 at 6:41
• Hmmm ... I think that a system of $N$ harmonic oscillators the ratio(s) of whose periods can not be expressed as rational numbers is ergodic. And that might contain a hint. Another parallel comes from the risks of using linear congruent PRNGs to draw higher dimensional tuples. – dmckee --- ex-moderator kitten Mar 6 '18 at 17:20

The orbits of the harmonic oscillator in 1D are closed curves in the phase space - and the key here is that these curves coincide with the (1D) energy surfaces $S$ of the system, which means that the energy surface is trivially fully explored by a trajectory (so the system may be ergodic).
In more dimensions, i.e., for two or more independent harmonic oscillators, the conserved quantities are the energies of the individual normal modes, defining a surface that doesn't coincide with $S$, which is the surface of total constant energy of the system. In other words, the system's conserved quantities are not constant in $S$, which therefore cannot be fully explored, which means the system cannot be ergodic.