Potential Energy Curve (stability)

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At a point where dU/dx = 0 and d²U/dx² > 0 the equilibrium is?

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1. **Equilibrium Condition**: For a conservative system, force $F = -\frac{dU}{dx}$. If $\frac{dU}{dx} = 0$, then $F = 0$, meaning the particle is in equilibrium.
2. **Type of Equilibrium**: This depends on the behavior of potential energy around that point.
3. **Stable Equilibrium**: Occurs at a **local minimum** of $U$. Mathematically, $\frac{dU}{dx} = 0$ and $\frac{d^2U}{dx^2} > 0$. Any small displacement results in a restoring force toward the center.
4. **Unstable Equilibrium**: Occurs at a local maximum ($\frac{d^2U}{dx^2} < 0$).
5. **Neutral Equilibrium**: Potential is flat ($\frac{d^2U}{dx^2} = 0$).

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Potential Energy Curve (stability)

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