Gravitational potential energy and continuous matter

[burakumin]

The gravitational potential energy of two massic points $P_1$ and $P_2$ with respective masses $m_1$ and $m_2$ is given by
$$U = -G \frac{m_1 m_2}{|| P_2 - P_1 ||}$$
Now I was wondering how this formula could be applied to continuous matter. Let us imagine a very simple case where we have a segment-like body (a cylinder with negligible radius) of length $L$ and constant linear density $\rho$. The gravitational potential energy of two infinitesimal segments centered in $x_1$ and $x_2$ (with $0 \leq x_1 < x_2 \leq R$) would be:

$$\delta U = -G \rho^2 \frac{\textrm dx_1 \textrm dx_2}{x_2 - x_1}$$

But now if we try to add every contribution:

$$U = -G \rho^2 \int_{x_1 = 0}^{x_2} \int_{x_2=0}^L \frac{\textrm dx_1 \textrm dx_2}{x_2 - x_1}$$

this integral diverges...

Of course real matter is not really continuous so a more relevant description of reality would rather be a finite sum of many close but not superposed massic points. However it seems quite unsatisfying that the standard formula of gravitational potential energy does not work with the very common assumption of continuous matter. Am I missing something ?

 

[jbriggs444]

I think the problem is that you have assumed that the cylinder is of negligible radius. But then you've taken a differential element that is, in the limit, smaller than any radius. Accordingly, the cylinder radius is not negligible.

 

[Dale]

For continuous media you will be better off using a differential expression rather than an integral expression. In this case, the best one is:
$\nabla^2 U = 4 \pi G \rho$

https://en.m.wikipedia.org/wiki/Gauss's_law_for_gravity