potential energy of a system of particles

[Rick16]

TL;DR

I have a conceptual problem with the potential energy of a system of particles

Here is problem 1.55 (a) from Schroeder's Thermal Physics:

Consider a system of just two particles, with identical masses, orbiting in circles about their center of mass. Show that the gravitational potential energy of this system is -2 times the total kinetic energy.

I did not know how to find the potential energy of this system, so I looked up the solution, according to which the potential energy of the system is simply the potential energy at the location of one of the particles. Why would that be? And what if I have a system with more than 2 particles? Would the potential energy of the system as a whole be the same as the potential energy at the position of one of the particles due to the presence of the other particles?

 

[A.T.]

I looked up the solution, according to which the potential energy of the system is simply the potential energy at the location of one of the particles

Can you post the exact wording of the solution?

 

[Rick16]

Can you post the exact wording of the solution?

Here is the solution:

The kinetic and potential energies of the system are $$U_k=2\cdot\frac{1}{2}mv^2=mv^2,~~~U_p=-\frac{Gm^2}{2r}.$$ To show how these are related, apply Newton's second law to the motion of one of the particles: $$F=ma\Rightarrow \frac{Gm^2}{(2r)^2}=m\frac{v^2}{r}.$$ Multiply each side of this equation by $2r$, and the left-hand side is the magnitude of the potential energy. Therefore, $$U_p=-2mv^2=-2U_k$$.

 

[Rick16]

One more comment to show where exactly my problem lies: Potential energy is defined at a specific position. A System does not have a specific position. How then can I define/understand what the potential energy of a system would be?

 

[PeroK]

One more comment to show where exactly my problem lies: Potential energy is defined at a specific position. A System does not have a specific position. How then can I define/understand what the potential energy of a system would be?

Potential energy is generally a function of the position of all particles in the system.