What is the correct gravitational potential energy of a single particle in an $N$-body system?

Question

I am aware that the total gravitational potential energy of a system of $N$ particles is given by pairwise interactions, i.e., you start with a single particle in the system, and then calculate the work done (negative for an attractive force) to bring in every other additional particle. Like this:

$$U_{total}=-G\sum_{i=1}^{N-1}\sum_{j=i+1}^{N}\frac{m_im_j}{r_{ij}}\tag{1}$$

However, does it make sense to talk about the gravitational potential energy of a single particle? Something like this:

$$U_i=-Gm_i\sum_{j=1,j\neq i}^{N}\frac{m_j}{r_{ij}}\tag{2}$$

However, as can be seen from equation 1, summing over these "individual" gravitational potential energies would result in pairwise interactions being counted twice. Thus, would this:

$$U_i=-Gm_i\frac{1}{2}\sum_{j=1,j\neq i}^{N}\frac{m_j}{r_{ij}}\tag{3}$$

... be a correct equation for the gravitational potential energy of the $i^{th}$ particle in an N-body system? At the very least, using equation 3 to calculate the potential energy of each particle would result in the correct total potential energy for the system when summing the inividual energies of the particles.

Any insight would be much appreciated.

Answer 1

The total potential energy of a system of $N$ particles interacting through Newton's gravitational force can always be written either as $$U_{total}=-G\sum_{i=1}^{N-1}\sum_{j=i+1}^{N}\frac{m_im_j}{r_{ij}}\tag{1}$$ or as $$U_{total}=-\frac{G}{2}\sum_{i=1}^{N}\sum_{j=1;j \neq i}^{N}\frac{m_im_j}{r_{ij}}\tag{2}.$$

In the form $(2)$, each pair of bodies is counted twice, then the necessity of a factor $1/2$.

Notice that $U_{total}$ is a property of the whole system. However, it is possible to interpret formula $(2)$ as the sum over all the bodies of the potential energy of one body at the time, in the presence of the remaining $N-1$. Therefore, your equation $(3)$.

Answer 2

The total potential energy of the whole system is not the sum of the individual potential energies of each particle while holding the other $n-1$ particles fixed. That’s because in order to build up the system, you need to bring the particles in one at a time. Or, more generally, the work required to bring in each particle only depends on the other particles that have already been brought in, but not on the ones that will be brought in later.

So a priori, one might expect that the total potential energy of the system might depend on the order in which the particles are brought in. But conveniently, it turns out that the order doesn’t matter; adding in the overall prefactor of $1/2$ takes care of avoiding double-counting particles, regardless of the order that they’re brought in.

So your equation (2) is the correct one, but it needs to properly interpreted. It only applies to the last particle that you bring in. (It can apply to every particle if you restrict the sum to only go over the particles that were brought in earlier than the particle in question, but that information is often not available, and fortunately we don’t need it.)

Answer 3

Equation (1) is relevant if you're studying the evolution of the whole system. For example, you can use it to construct the conserved energy or the Lagrangian.

Equation (2) is relevant if you're studying the kinematics of the subject particle over a short time period (so you neglect motion of the rest of the system). For example, its gradient (with respect to $\vec r_i$) tells you the instantaneous force on your particle. This equation might be used when numerically integrating the system's evolution.

I can't think of a context where equation (3) is useful, except as a computation step toward evaluating equation (1).