Conservative Forces and Potential Energy

If the force is conservative, i.e. $\vec{F} (\vec{r}) = - \nabla U (\vec{r})$ , then:

W = \int_{1}^{2} (- \nabla U) \cdot d \vec{r} = - (U_{2} - U_{1})

Thus, conservative forces have an associated potential energy function $U$ . For conservative forces, we see that work $W$ is path-independent, depending only on end-point potential energies:

W = K_{2} - K_{1} = - (U_{2} - U_{1})

K_{1} + U_{1} = K_{2} + U_{2}