How to characterize a power law field?

[Moleculmol]

Hello, as part of the study of fields with central forces, I came across with fields called power law, defined by F = - K/r ^ n u
(u is radial vector passing through the origin O)
I would like to dismiss case n = 2, which refers to the Newtonian fields whose study was exhaustively conducted in class. My questions are:
-How to recognize a power law field from the trajectory of a moving material point in this field?
- Once the nature of the field identified (power law), the determination of the power n can be conjectured by reasoning on heuristic considerations, however are there any situations where one can resort to theoretical models to find this coefficient (I think for example to VanderWalls force but I would like another example).
- An example of application of these methods to a concrete physical situation, in cosmology (movement of the stars) or interactions between particles at infinitesimal scale.
Thank you in advance.

 

[mfb]

Summary: In classic mechanics, central forces with the form -k/r^n are often introduced to model actions that are directed along the line joining the object and the origin.

-How to recognize a power law field from the trajectory of a moving material point in this field?

Calculate how the motion will look like for different powers, then do the opposite if you have a given trajectory. n=2 and n=-1 are the only cases that lead to closed non-circular orbits.

- An example of application of these methods to a concrete physical situation, in cosmology (movement of the stars) or interactions between particles at infinitesimal scale.

Well, on large scales these are all n=2, on small scales you probably have to consider quantum mechanics and things get more complicated.