Harmonic Oscillator - Application To A Conservative Force

Application To A Conservative Force

The problem of the simple harmonic oscillator occurs frequently in physics, because a mass at equilibrium under the influence of any conservative force, in the limit of small motions, behaves as a simple harmonic oscillator.

A conservative force is one that has a potential energy function. The potential energy function of a harmonic oscillator is:

Given an arbitrary potential energy function, one can do a Taylor expansion in terms of around an energy minimum to model the behavior of small perturbations from equilibrium.

Because is a minimum, the first derivative evaluated at must be zero, so the linear term drops out:

The constant term V(x₀) is arbitrary and thus may be dropped, and a coordinate transformation allows the form of the simple harmonic oscillator to be retrieved:

Thus, given an arbitrary potential energy function with a non-vanishing second derivative, one can use the solution to the simple harmonic oscillator to provide an approximate solution for small perturbations around the equilibrium point.