Small oscillations (potential energy)

A conservative force can be related to potential energy $V$ by $F=-\frac{d}{dx} V(x)$ . If the potential energy curve at a point of equilibrium $x_0$ (i.e. $V(x_0)=0$ ) is even, a particle may oscillate around $x_0$ .

In this case, we can approximate the force acting on the particle using a Taylor expansion:

V(x) = V(x_0) + \frac{V'(x_0)}{1!}x + \frac{V''(x_0)}{2!}x^2 + \cdots.

\begin{align*} F(x) &= -\frac{d}{dx} V(x) = V'(x_0) + \frac{V''(x_0)}{1!}x + \frac{V'''(x_0)}{2!}x^2 + \cdots \\ &\approx V''(x_0)x. \end{align*}

This approximation is good as long as $V''(x_0) \gg \frac{V'''(x_0)}{2}x$ . Since this approximation is linear we can solve for $x$ the same way as a regular spring equation using Hooke’s law.