# Why a system should be at its lowest energy state for its stability?

To answer your question, you should first understand when is a system ** most stable**.

Firstly it shouldn't have a tendency to move or change state, thus it should be under equilibrium conditions, i.e. the net Force should be zero.

We know that $$F = - \frac{dU}{dx}$$ Putting $F=0$, we get $$\frac{dU}{dx}=0 \tag{1}$$

Secondly, it should be able to maintain that equilibrium condition by itself. This can be tested by displacing the system by a small distance $\delta x$. If the force on the system then becomes opposite to direction of $\delta x$, we can say that the system has a tendency to restore back to its original equilibrium position.

An example of this would be a ball kept at the bottom of a spherical valley. Displace the ball a little towards the right, and the net force on it acts towards left, bringing it back to its original position. You will realise that I just described a stable equilibrium condition. What this proves is that it is the stable equilibrium condition in which the system is ** most stable**.

From the above description we have that the small displacement $\delta x$ and net extra force $\delta F$ should be in opposite directions

$$\delta F = \frac{dF}{dx} \delta x + \mathcal{O}(\delta x^2) \approx \frac{dF}{dx} \delta x$$

which gives as stability condition

$$\frac{dF}{dx} < 0$$

which implies

$$-\frac{d^2U}{dx^2}<0$$ $$\frac{d^2U}{dx^2}>0\tag{2}$$

From $(1)$ and $(2)$ it is evident that the graph of $U$ should have a minima at stable equilibrium condition, i.e. **The Potential Energy should be minimum when a system attains maximum stability.**