# Optimal mass distribution for maximal gravitational field

1. ### Heirot

151
1. The problem statement, all variables and given/known data

Suppose that one is given a lump of clay of total mass M and constant density. Let P denote a particular point in space. In what way should one shape and position the clay so that the gravitational field in point P is maximum possible? It is assumed that the clay stays in one piece during the shaping.

2. Relevant equations

Newton's law of gravity
Variational principle (?)

3. The attempt at a solution

I don't know if this problem has an obvious and trivial solution, but I'm thinking along the lines of variational principle. Let P be the origin of coordinates with z axis pointing in the direction of the gravitational field. Then we have the following equations

$$F=G \rho\int_{\mathcal{V}}dV\frac{\cos \theta}{r^2}$$

$$M=\rho V$$

Can I maximize the force by varying the volume of integration using the second equation as a constraint?