# Potential Energy in a System and its Relation to the Total Energy

In this answer it was stated that potential energy is a property of a system and not an individual particle.

If we have two particles (1 and 2) interacting via a conservative force, we can write an equation regarding the total energy of the system and its conservation:

KE1 + KE2 + PE(sys) = KE1' + KE2' + PE(sys)'

Where the ' indicates final.

I know this equation is incorrect, but my question is why is it incorrect?

I most definitely am interpreting this answer incorrectly.

• When one part of the system is effectively invariant---as in the case where human sized things are moving around near the surface of the Earth---it is expedient to ascribe the potential energy to the small thing. It's just that if someone took the planet away it would affect the behavior of the experiment. On the other hand, if you are talking about the elastic energy of a spring of natural length $l_0$ with a mobile mass on each end, then the potential should be written $U(\vec{x}_1,\vec{x}_2) = \frac{1}{2}k(|\vec{x}_1-\vec{x_2}|-l_0)^2$ and is obviously a function of both positions. – dmckee --- ex-moderator kitten Oct 9 '15 at 2:20
• Your equation is correct. Why do you think that its incorrect? – garyp Oct 9 '15 at 12:56
• @garyp So when finding the total energy of the system it would be correct to count the potential energy once (instead of twice for both particles)? – andrew Oct 9 '15 at 21:16
• Potential energy is the energy associated with the interaction between pairs of objects, and the positions of those objects. So you calculate it once for every pair of objects. Two objects: once. Three objects: three times (and add them together). Four particles: six times (and add them together). One particle: not defined! If you throw a ball, there are two objects interacting by gravity: the ball and the earth. – garyp Oct 9 '15 at 22:45
• @garyp You can derive KE1+PE(sys)=KE1'+PE(sys)' from the work-energy theorem and the definition of potential energy. Doesn't this contradict KE1 + KE2 + PE(sys) = KE1' + KE2' + PE(sys)'? I'm still a bit unclear on this. Could you provide any references on this subject? Thanks. – andrew Oct 10 '15 at 4:50