# Reverse engineering of particle motion

Is there an algorithm that can determine the interaction laws between particles (or point-like objects) based on their motion?

A simple case: an algorithm that determines the gravitational interaction from nothing but the motion of the planets in the solar system. Without a lookup table of particle interactions.

• Topological sensitivity analysis is used in biological models to determine the interactions between genes in a gene regulatory network. Maybe that could be adapted to your domain? Aug 29 '17 at 20:38

If you make the assumption that the force the two objects exert on each other is along the connecting line (this is necessary for the conservation of angular momentum, but assumes an infinite speed with which forces penetrate space), then you have that the differential equation for particle 2 is $$\ddot x_2(t) = f(\|x_2(t)-x_1(t)\|) (x_2(t)-x_1(t))$$ and similarly for $x_1(t)$. If you know the trajectories of both particles, then you can compute $$f(\|x_2(t)-x_1(t)\|) = \frac{\|\ddot x_2(t)\|}{\|x_2(t)-x_1(t)\|}$$ and you can tabulate this force $f(r)$ for all values of $r$ that appear as distances $r=\|x_2(t)-x_1(t)\|$ over the course of the observation period. At least for this range of $r$ values, you then know what the interaction force is.