  It is shown that a spin system is equivalent to a set of constrained harmonic
oscillators. For finite, but large, systems, a continuous approximation to the
density of states can be used, and the oscillator frequencies can be exactly
computed. In the phase transition, the effective frequency of the lowest mode
passes through zero, that is, it becomes an inverted oscillator. In the small
oscillations regime, the oscillators can be treated as independent and the
thermodynamic magnitudes can be computed. We show explicit calculations in a
disordered, frustrated, high coordination number Blume Capel model with 60,000
spins.
