Potential Energy

Consider a particle subject to a conservative force. Let $O$ be the origin of our coordinate system (i.e., the point whose displacement is ${\bf0}$), and let $P$ be a general point whose displacement is ${\bf r}$. We can define the function

$$U({\bf r}) = -\int_{\bf0}^{\bf r} {\bf f}({\bf r}')\cdot d{\bf r}'.$$ (1.41)

The fact that the force is conservative ensures that this function has a unique value at each point in space. On the other hand, if the force were non-conservative then the function would be ill-defined, because there are an infinite number of different paths linking points $O$ and $P$, and each path would yield a different value of the integral on the right-hand side of the previous equation. The quantity $U$ is known as potential energy, and is the energy that the particle possesses by virtue of its position. Obviously, it only makes sense to associate potential energy with a conservative force. Note that the fact that the position of the origin of our coordinate system is arbitrary implies that potential energy is undefined to an arbitrary additive constant. In other words, only differences in potential energies are physically meaningful.

Suppose that the particle moves from point ${\bf r}$ to point ${\bf r}+d{\bf r}$. The associated change in the particle's potential energy is

$$dU =- {\bf f}\cdot{\bf r}= - f_x\,dx-f_y\,dy-f_z\,dz.$$ (1.42)

Suppose that $dy=dz=0$. We can write

$$f_x = - \left(\frac{dU}{dx}\right)_{{\rm constant}\, y,\,z} =- \frac{\partial U}{\partial x}.$$ (1.43)

Similar arguments yield

$$f_y = - \frac{\partial U}{\partial y},$$ (1.44)

$$f_z =-\frac{\partial U}{\partial z}.$$ (1.45)

Hence, we deduce that

$${\bf f} = -\frac{\partial U}{\partial x}\,{\bf e}_x -\frac{\partial U}{\partial y}\,{\bf e}_y-\frac{\partial U}{\partial z}\,{\bf e}_z,$$ (1.46)

where ${\bf e}_x$ is a unit vector parallel to the $x$-axis, et cetera. (See Section A.4.) The previous equation can be written more succinctly as

$${\bf f} = -\nabla U.$$ (1.47)

(See Section A.19.) In other words, a particle moving in a conservative force field experiences a force that is equal to minus the gradient of the potential energy.