Equilibrium

Recall that the force is given by the negative gradient of the potential energy:

$$\vec{F} = -\vec{\nabla} U$$ (3.48)

or (in each direction)

$$F_x = -\frac{dU}{dx}$$ (3.49)

or the force is the negative slope of the potential energy function in this direction. We can use this to rapidly evaluate the behavior of a system on a qualitative basis.

Consider the following potential energy curves:

In this first one, the slope is negative so $F_x$ is positive. Near the origin the force is large, and farther out it gets weaker. This is a ``repulsive'' potential energy function.

This is a more useful one. This curve is typical of a ``molecular potential''. At short ranges it is strongly repulsive. There is a minimum that corresponds to a stable equilibrium point of separation. There is an unstable equilibrium point somewhat farther out (where $\vec{F} = 0$ but where any motion around this point will cause the mass to ``fall away'' to infinity or into the attractive potential). Beyond this the force points out.

Clearly, the force points the way we intuitively expect things to ``slide'' down these potential energy curves. In fact, as something moves into a region of lower potential energy, its kinetic energy must increase, so that its velocity increases. This means that there is a force pointing in the direction of the potential energy's decrease, which is what we mean when we write the gradient relation above.