Editing Harmonic oscillator (section)

== 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 is associated with a [[potential energy]].  The potential-energy function of a harmonic oscillator is
<math display="block">V(x) = \tfrac{1}{2} k x^2.</math>

Given an arbitrary potential-energy function <math>V(x)</math>, one can do a [[Taylor series|Taylor expansion]] in terms of <math>x</math> around an energy minimum (<math>x = x_0</math>) to model the behavior of small perturbations from equilibrium.

<math display="block">V(x) = V(x_0) + V'(x_0) \cdot (x - x_0) + \tfrac{1}{2} V''(x_0) \cdot (x - x_0)^2 + O(x - x_0)^3.</math>

Because <math>V(x_0)</math> is a minimum, the first derivative evaluated at <math>x_0</math> must be zero, so the linear term drops out:
<math display="block">V(x) = V(x_0) + \tfrac{1}{2} V''(x_0) \cdot (x - x_0)^2 + O(x - x_0)^3.</math>

The [[constant term]] {{math|''V''(''x''<sub>0</sub>)}} is arbitrary and thus may be dropped, and a coordinate transformation allows the form of the simple harmonic oscillator to be retrieved:
<math display="block">V(x) \approx \tfrac{1}{2} V''(0) \cdot x^2 = \tfrac{1}{2} k x^2.</math>

Thus, given an arbitrary potential-energy function <math>V(x)</math> 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.