On the normal modes of coupled harmonic oscillators

We discuss in detail a well known method for obtaining the frequencies of the normal modes of coupled harmonic oscillators that is based on the simultaneous diagonalization of two symmetric matrices. We apply it to some simple illustrative examples recently chosen for the presentation of an alternative approach based on a set of constants of the motion. We show that the traditional method is simpler, even in the case of equal frequencies. We also discuss a problem that commonly appears in elementary courses on quantum mechanics that also requires the diagonalization of two symmetric matrices.