The pair potential different that $r^{-1}$

Question

As I know, the potential between two particles of the form ~ $r^{-1}$ ($r$ is distance between particles) is special, because it solves the Poisson's equation in 3D.
My question is: If I consider for example pair potentials like $r^{-2}$, or $\ln(r)$, can I construct linear equations for this fields, equivalent to Poisson equation for gravitational potential?

Answer 1

There's few mistakes you've made.

1. If you take E&M you would knew that the solution came with basis in $r^a$ where $a$ to be integers.

2. For solutions of specific question(such as comet or charges in free space, if I remembered correctly), sometimes $r^{-1}$ marked a "tipping-point", of which for $a<-1$ they($r^a$s) usually satisfy some nice properties. However, if the question get changed, the specialty of $r^{-1}$ may no longer holds.

3. Usually, one don't need to consider $ln(r)$ when using basis such as $r^{a}$s.

For your question, most times in classical or E&M was yes, because most of the conservative quantity you mentioned, such as "energy", was additive, and satisfied the linear relationships. However, if the quantity was not additive, such as the square root, then your question is false.