Is there always an equilibrium point in a field?

Question

For instance, considering a set of planets represented as point masses that create a gravitational field, will there always, no matter what set of points, be a place where I can stand with no net force acting upon me? For a simple two or three body case, there seems to always be a point or a region, but I'm not sure if it generalizes to all cases.

Answer 1

It depends. As Robert points out in the comments, there are trivial cases where there are no such points, but in some cases (more than one point mass), there always must be.

Mathematically speaking, what you are asking is the following: given the gravitational potential $V(x)$ due to some set of massive bodies, does there exist any point $x$ where the force $-\nabla x =0 $? In other words, does $V$ have any critical points?

I'm assuming you do not consider a point occupied by a body to be an equilibrium point. In that case, the shape of $V(x)$ is very special: because the gravitational potential is harmonic, $V$ satisfies the maximum principle: $V$ cannot have any local maxima or minima except at mass points. So any equilibrium points $\nabla V = 0$ must be saddle points.

Is it possible that $V$ has no saddle points? Well yes... if you have single body. What about if you have multiple bodies? The simplest case is if you restrict yourself to point particles. Then look at the level sets of $V$: for values of $V$ close to zero, the level set looks like a single closed surface with the topology of a sphere, approximating a sphere of very large radius about the center of mass of the points. As $V$ goes to negative infinity, the level sets looks like many tiny spheres, one around each mass point. At some values of $V$, the single large sphere must split into several smaller spheres, and at these places where the level sets change topology, you must have saddle points.

What if your bodies aren't point masses? Thinks become more complicated. You can construct examples where you have multiple bodies but no equilibrium points. Perhaps it's easiest to think in 2D: consider two infinite lines, one $y=1$ with density 2 kg/m and the other $y=-1$ with density 1 kg/m. It is an easy exercise to show that the gravitational potential is piecewise linear in $y$ (looks something like this) with a global minimum along the top line and no saddle points.