I want to study the evolution of a particle as a function of time, then in dynamical systems, the usual thing to do would be to define the state of the particle. Usually we are able to do this by doing a linear transformation.
Now, let us say that two particles now exist. It is possible that the state of each particle can be studied 'independently' if the particles do not interact at all.
My question is, what if the particles interact with each other? (That is, some sort of information gathering is made from one particle to another so that these particles are not independent from each other?)
What other forms/types of models exist (or what modifications do I make to the usual state representation above) to accommodate these changes?
Your insights will be helpful.
Generally you write a set of coupled differential equations for the locations of the particles. Each one may be reacting to a potential, but there is also an interaction force. If $a$ is the position of the first particle, you have an equation like $a''=f(a)+g(a-b)$ where $f$ represents the potential and $g$ is the interaction force. You have a similar equation for the second particle. In 3D you now have six differential equations, one for the acceleration of each particle in each axis.