definition of potential energy

So now we sorted out the idea of conservative forces, we can define the potential energy. Potential energy is a function of coordinates so you could write it as $U(x,y,z) ~=~ U({\bf r})$. One thing about potential energy is that it's kind of like when you define a coordinate system: you get to choose where you'd like it to be zero. In other words there is no unique definition of potential energy, you can add any constant to it that you like. Note that it doesn't make it meaningless at all, it is a very useful concept. It's just like when you take an anti-derivative, it includes an additive constant. So we'll define the potential energy difference between two arbitrary points ${\bf r}_i$ and ${\bf r}_f$.

$$\Delta U({\bf r}_f) ~=~ -W ~=~ - \int_{{\bf r}_i}^{{\bf r}_f} {\bf F}\cdot d{\bf r}$$ (1.16)

the last equality using the definition of three dimensional work eq. 1.15.

So what do we do about this darned additive constant? Well it's actually quite useful. Since we can set it to be anything we like, we can choose it to be zero at a particularly convenient point. So if we said that at $U ~=~ 0$ when ${\bf r}={\bf r}_i$ then we'd have $U({\bf r}) ~=~ - \int_{{\bf r}_i}^{{\bf r}} {\bf F}\cdot d{\bf r}$.

Now you might say that this definition was idiotic because I haven't specified the path to take when doing the line integral. That's where the fact that the force must be conservative comes into the definition of potential energy. Yes, potential energy is not possible to define if you have a non-conservative force like friction say.