Potential Energy Defined
In all cases when we deal with a conservative vector field such as with gravity, we can do our accounting of energy differently to ultimately save ourselves time. As you will see in this section, that is the whole point of potential energy. It is not some new type of energy in any fundamental sense.
In the last chapter we attributed to a single mass (like a ball) only rest energy and kinetic energy In that case, when dropped in a gravitational field, the ball had work done on it by gravity as it fell. Gravitation, and the earth which is responsible for that gravity, was seen as external to the ball in the sense that the ball was our system and the earth was acting on our system. In other words, gravity was just some outside force that did work on the ball.
We saw, that in spite of an infinite number of different paths that a ball can take from a higher to a lower height, that the work done by gravity (or any other conservative vector field) is always the same. Since that's the case it doesn't make sense to calculate work using a path integral each time we start a new problem since the path does not matter. For conservative vector fields, we should instead calculate it once, using the simplest choice of path (since it doesn't matter), and then use the result of the integral in our subsequent calculations. This is where potential energy is born... by the renaming of the work done by a conservative force. Please re-read that last part if you didn't absorb it. It is the essence of this chapter, and without getting it you will not understand the topic very well.
Let's look at the mathematics of this renaming of work now.
Inventing Potential Energy
Let's look at how we arrive at potential energy by starting with our work - kinetic energy principle from last chapter,
I will separate on the left side, the work due to non-conservative (NC) forces from the work done by conservative (C) forces like gravity. The reason it makes sense to separate these terms is that work done by conservative forces doesn't depend on the path (particulars of a problem), and work done by non-conservative forces DOES depend on path. So it is always necessary to calculate the path integrals to find work done by non-conservative forces... there is no short cut. Separating the conservative from non-conservative gives:
The results of the conservative path integrals only depend on the end points of the motion and not on the path chosen. Therefore I will put them on the right side of the equation along with the kinetic energy terms which also only depends on the end points (initial and final velocities). Then we just rename the conservative work and call it the negative of potential energy. This leads to:
The last line of the math is the definition of potential energy. If we contrast last chapter and this chapter, in both cases we see a common message: Work changes the energy of a system. The only issue is that we have a choice of perspective. Last chapter our system was always a single, isolated mass being acted on by the outside world. That outside world did work to change the kinetic energy of the mass. This could be seen as:
The perspective of this chapter is different. The only "outside" forces are those produced by non-conservative forces or vector fields like the wind blowing, the force of air drag, sliding friction, etc. The right side of the equation contains new energy terms associated with our system. When we find a new term called potential energy due to gravity on the right side of the equation, we should understand that now the earth (source of the gravity) is seen as part of our system.
Contrast
I want to be clear that you see how this new picture is different than last chapter. In the last chapter we found, for instance, that as a ball falls it gains kinetic energy. The gain in kinetic energy was from work done by gravity. In that view, earth (the source of the gravity) is external to the ball.
In this chapter we include the earth as part of the system. So now, it's a ball-earth system that has an associated potential energy. Being part of the system, earth's gravity will not do work on the system. Instead, as the ball gains kinetic energy while falling, the earth/ball system's potential energy is decreasing. It's a transfer of energy (potential to kinetic) within the system now. If the ball gains 10 joules of kinetic energy while falling, the earth/ball system will consequently lose 10 joules of potential energy. We instead just assume that energy is changing forms - from potential to kinetic. In this picture, work only needs to be considered if a non-conservative force acts on the ball in addition to gravity.