Nature of stability in a conservative system

It is a rather fundamental mathematic concept. The roots of first derivative are points of local extrema, where forces are vanishes. But these extrema could be local minimum (a stable equilibriium point), or local maximum (a unstable equlibrium point). The criterion of checking local maximum or minimum or saddle point (in higher dimension space) is to check the second derivatives.

Let use Taylor expansion to illustrate the potential near an equilibrium point $x_0$:

$$ V(x) = V(x_0) + 0 (x - x_0) + a (x - x_0)^2 + O(Delta x)^3; tag{1} text{ where the coefficient } a = frac{d^2V}{dx^2}|_{x=x_0}. $$

The force near $x_0$ is $$ F(x) = -frac{dV}{dx} = - 2 a (x - x_0). tag{2} $$

From Eq.(2), the force resembles a restoring force of simple harmonic oscillation if $a > 0$, and $x_0$ is the equilibrium point.

When $a < 0$, the force is positive pushing away from the equilibrium points.

From mathematic point of view,, the $frac{d^2V}{dx^2}|_{x=x_0} > 0$, the curve has a positive curvature. The potetial curve has a local minimum at $x_0$, and potential bends upwards in both side. On the other hand, $frac{d^2V}{dx^2}|_{x=x_0} < 0$, the potential has a local maximum at $x_0$. When $frac{d^2V}{dx^2}|_{x=x_0} = 0$, it is a turning point.