Potential Energy

Definition: Potential Energy

A potential energy function $U$ of a conservative force $\mathbf{F}$ is a scalar field such that

$\mathbf{F}=-\nabla U$

There are infinitely many such functions but we are free to choose where $U=0$ and thus fix a particular one, so long as we are consistent with our choice. Once $U$ is fixed, we call $U(\mathbf{r})$ the potential energy of a point mass at the position $\mathbf{r}$ due to $\mathbf{F}$.

NOTATION

$E_{\text{p}}{E}_{\text{pot}}U$

Theorem: Work of a Conservative Force

The work done by a conservative force $\mathbf{F}$ acting on a physical system is equal to the negative change in its potential energy.

$$W=-\Delta U$$

NOTE

The exact choice of the potential energy function is irrelevant.

PROOF

$W={\int }_{t_1}^{t_2}\mathbf{F}\cdot \mathbf{v}\mathrm{dt}=-U(\mathbf{r}(t_2))-(-U(\mathbf{r}(t_1)))=-U(\mathbf{r}(t_2))+U(\mathbf{r}(t_1))=-\Delta U$