My interpretation is now as follows. We first define our system. Then we have these equations: Work

_{total}= ΔK

_{system}, Work

_{external}= ΔE

_{system}, -1*Work

_{internal, conservative}= ΔU

_{system}, and Work

_{external}+ Work

_{internal, non-conservative}= ΔE

_{mechanical}. Conservative forces have to be internal to our system to define potential energy. Also, the potential energy we define is of our total system, not divided among the components of our system in some manner. Does anything seem wrong with my interpretation so far?

I don't want this initial post to get too long, so I'll cut to some questions I have:

**1)**We construct potential energy functions from the equation ΔU

_{system}= -1*Work

_{internal, conservative}. Let's talk about gravitational potential energy in a system where m1 << m2, such as a ball-earth system. We have

*two*internal, conservative forces in this system, the equal and opposite gravity forces. However, as the earth is much more massive than the ball, the gravity force by the ball on the earth essentially does no work on the earth; the earth isn't going to experience much of a ΔK ∴ the resulting -ΔU contribution from the earth's ΔK will also be negligible. Is this what leads to our approximation that "the potential energy of the ball-earth system" = "the potential energy of the ball"?

**2)**What about in other systems where we shouldn't ignore the second, third, etc. internal conservative forces? Take a horizontal spring-mass system. Why is it that we ignore the equal and opposite force that the mass exerts on the end of the spring? In our potential energy function calculation, we only consider minus the work of the spring force onto the mass.

**3)**Plenty of sources state that electric potential is the potential energy per unit charge that a test charge will possess at some point in space (relative to an arbitrary origin of course). However, doesn't this immediately contradict the notion of potential energy belonging to a system and not one particle (in this case the test charge)? I'm thinking that something similar to

**1)**is going on, perhaps because we're dealing with an electrostatic field.