Concrete example of a two-dimensional harmonic oscillator I am a student of mathematics and some time ago I showed in general that for a two-dimensional harmonic oscillator one can apply the recurrence theorem. So far so good.. now I would like to have a concrete example of a two-dimensional harmonic oscillator. Can you giuve me one?
 A: The harmonic oscillator is so incredibly important in physics as a whole because of the following consideration. For a more or less smooth potential $V(x)$ with a local minimum at position $x_0$, one can taylor-expand around that minimum:
$$V(x)\approx V(x_0)+\frac{V''(x_0)}{2}(x-x_0)^2+\mathcal{O}((x-x_0)^3).$$
The linear term is zero, otherwise it wouldn't be a minimum. Also, the coefficient in front of the quadratic term has to be positive for the same reason. Often, potentials are symmetric around local minima due to the symmetry of the problem, in which case all the odd-power terms will vanish as a rule, so the approximation is even better. The pendulum mentioned in the comments is an example for this. There, the potential has a cosine dependence on the displacement.
Due to energy considerations it is immediately clear that the region of applicability of the harmonic approximation (that's what it's called by the way) will not be left. Therefore, it is safe to replace the full potential by the quadratic ("harmonic") approximation.
For the desired two-dimensional harmonic oscillator, the argument is exactly the same. If it is supposed to be isotropic, meaning that the restoring force's magnitude only depends on the distance from the equilibrium point, not on the direction, the potential is rotationally symmetric around it's minimum (at least up to second order).
Exact harmonic oscillators are quite rare in physics. They're almost always approximations following the above scheme.
