One dimensional potential energy

Potential energy is such a gosh-darned useful concept that many times it's easier to think about physics in terms of it, rather than forces. We'll try to illustrate this in one dimension and use potential energy to describe the ``stability of equilibria''. So if now we want to think in terms of potential energy $U(x)$, we would like to know how to relate the potential energy of an object, to the force acting on it.

Using eqs. 1.6 and 1.15 we see that in one dimension

$$U(x) ~=~ -\int_{x_i}^x f(x) dx + U(x_i)$$ (1.31)

We can choose $U(x_i)$ arbitrarily, so basically what this equation says is that $U(x)$ is the anti-derivative of $-f(x)$. That means that

$$f(x) ~=~ -{dU\over dx}$$ (1.32)