Physics

Equilibrium happens where the force is zero—that is, where the potential energy stops changing, or $\frac{dU}{dx} = 0$. The key insight is that stability depends on the shape of the potential curve at that point: if the potential curves upward (second derivative positive), the equilibrium is stable, like a ball at the bottom of a bowl. Find where the slope is zero, then check the curvature to confirm it's a minimum, not a maximum.

Equilibrium happens where the force is zero—that is, where the potential energy stops changing, or $\frac{dU}{dx} = 0$. The key insight is that stability depends on the shape of the potential curve at that point: if the potential curves upward (second derivative positive), the equilibrium is stable, like a ball at the bottom of a bowl. Find where the slope is zero, then check the curvature to confirm it's a minimum, not a maximum.