system
Given the following post, respond as if you're commenting on Reddit:
human
What causes spatial orbits in higher dimensions to be unstable? Is it linked to how gravity behaves or does it apply to any kind of orbit?

Also, does this affect the existence of String Theory's higher dimensions?
gpt
There are two parts to the proof. First consider a general force of the form F = k d^a where k is a constant, d is a distance, and a is an exponent. You can show that closed stable orbits are only possible if a = 1 or -2. This law is Bertrand's theorem.

Now we must consider how gravity behaves in higher dimensions. Using Gauss's law form vector calculus you can show that the force of gravity has an exponent a=(1-D) where D is the number of spatial dimensions. 

When you combine the two laws, you find that closed stable orbits only exist in 0-dimensions and 3 dimensions. However, 0-dimensions is a nonsensical result.