Advantage of Poisson's equation?

[peter46464]

What's the advantage of expressing Newton's universal law of gravitation in terms of Poisson's equation? I read that Poisson's equation is important but why is this? What can you do with it that you can't do with Newton's inverse square law?

Thank you.

 

[Born2bwire]

The Newtonian law is one for a point like mass or for the case where we have a spherical mass which we are looking at points exterior to. The Poisson equation is much more general because it allows us to describe the gravitational potential due to a distribution of mass. We can think of using to find the gravitational fields inside a spherical shell or due to a non-spherical object. One could use the Newtonian equation as a Green's function to solve these problems too but it may be more cumbersome than solving a differential equation.

 

[peter46464]

Thank you. So if, for an esoteric example, there was an iron, cube-shaped planet, could you use Poisson's equation to find the gravitational field inside, outside and on its surface? I thought the equation only provided an exterior solution?

 

[Born2bwire]

The equation is valid throughout the volume of space. It can be fairly easy to use it to find exterior solutions for some problems (i.e., the aforementioned spherical mass) but if you model the mass as a density function then you can find interior solutions as well. So yes, you could use Poisson's Equation as a basis for solving the gravitational field of a cube shaped planet. It would probably be very difficult to solve it in closed form (if such a solution exists), but you could easily use Poisson's equation as your starting point for a numerical solution.