I don't know how the gravitational potential energy of a system of particles is defined. For example, how would one calculate the potential energy of a system of two stars, one mass $M$, the other mass $3M$, seperated by a distance $d$.
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2 Answers
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This is a addendum of @JgL answer. This is easily generalizable for an $n$-body system: $$ V = -\sum_{i<j}\frac{Gm_im_j}{|\mathbf r_i - \mathbf r_j|} $$
answered Jan 8, 2015 at 2:36
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In classical Newtonian gravity the potential energy of a system is defined as $$V = -\frac{GM_1 M_2}{r} = -\frac{G3M^2}{d}$$ This is the amount of energy you need (or in this gain will gain) when you pull a mass $m$ closer to another mass of $M$ starting with the two masses at infinite separation. Thus, it is equal to the integral $$V(r) = -\int_r^\infty \vec{\mathbf{F}} \cdot d\vec{\mathbf{r}} = -\int_r^\infty \frac{-GmM}{r'^2} \cdot dr' = -\frac{GmM}{r}$$
