Newton's Law of Universal Gravitation for extended objects

Carroll and Ostlie write, in their book An Introduction to Modern Astrophysics (second edition), at the page 33, right after deriving Newton's Law of Universal Gravitation for two point masses using Kepler's third law and the three laws of motion :

Newton's law of gravity applies to any two objects having mass. In particular, for an extended object (as opposed to a point mass), the force exerted by that object on another extended object may be found by integrating over each of their mass distributions.

My question is : how is the integrating process a "particular" case of the formula for two point masses ? Shouldn't the integrating process really be a postulate on its own ?

• The integrating process is a case of the two-point-mass situation since when you integrate, you are integrating over pairs of volume elements $dV_1$ and $dV_2$, with the idea that each pair of volume elements can be considered a pair of pointwise particle attractions. By adding up all the pairs, you get the total force. – DumpsterDoofus Mar 19 '14 at 23:40
• Okay, but what enables you to "consider" the two volume elements as being point masses ? Isn't this a postulate ? The integration is a limiting process and in no way during the process are we treating point masses, are we ? – Guest Mar 19 '14 at 23:43
• Yes, in the limiting process you are considering point masses, because the size of the volume elements approaches zero, aka point masses. – DumpsterDoofus Mar 19 '14 at 23:57
• I still maintain that while the volume elements approach zero, they are never zero during the limit process... – Guest Mar 19 '14 at 23:59
• That seems like a physically unimportant distinction, especially considering that you could alternately choose to discretize the object as a grid of point particles, rather than cubic volume elements. Or, if you're really looking for an axiomatic approach, you could take the integral definition as being a fundamental axiom for calculating forces, whereupon the point particle formula for attractive force is naturally derived by modeling them as Dirac delta functions. – DumpsterDoofus Mar 20 '14 at 0:16

If you are really concerned about an axiomatic treatment, here is a method that takes the integral definition as a postulate, and then recovers the point mass situation as a special case: define the "interaction energy" of a pair of nonoverlapping mass distributions $\rho_1,\rho_2$ as $$U=G\int_{V_1}\int_{V_2}\frac{\rho_1(\mathbf{r}_1)\rho_2(\mathbf{r}_2)}{|\mathbf{r}_1-\mathbf{r}_2|}$$ where $V_1$ and $V_2$ are the relevant spatial extents of $\rho_1$ and $\rho_2$. While it's not immediately obvious that this definition is "correct" (although it does have the correct units of energy), it reproduces the experimentally observed behavior of gravitational interactions between objects. In particular, for the case of point masses $\rho_1(\mathbf{r})=m_1\delta(\mathbf{r}-\mathbf{a}),\rho_2(\mathbf{r})=m_2\delta(\mathbf{r}-\mathbf{b})$ one obtains (by integrating) $$U=\frac{Gm_1m_2}{|\mathbf{a}-\mathbf{b}|}$$ which is exactly the potential form of Newton's law of gravitation for a pair of point masses (to get the more common force form, differentiate $U$ with respect to $(\mathbf{a},\mathbf{b})$).