Here's the problem from the homework. I've called the initial positions in order as 0, l, and 2l.
The most important equation here would have to be
|V - w2*M| = 0,
where V is the matrix detailing the potential of the system and M as the "masses" of the system. This eigenvalue equation would normally be solved after getting a V and M to find the eigenfrequencies, and the normal modes from there. However, this problem is asking for the individual positions, so I'm guessing normal modes isn't the way to go unless we are to parse out the equations for the individual positions from the normal modes.
Another equation is E = -grad(V), since we could get the electric potential from our electric field and from that, the electrical potential energy depending on the position of the particles.
However, I don't feel like the eigenvalue equation will help much since the potential is time dependent. Maybe I'm wrong about this.
The Attempt at a Solution
My first attempt was to just write out the potential based on the springs and the electric field. However, I realized the issue with this method is that the time dependence of the electric field will complicate things, and I'll get the normal modes instead of the individual positions.
My next attempt was to write out Newton's equations for each of the individual particles, but these equations turned out to be too complicated to solve in a straightforward manner.
I'm wondering what the best way to proceed on this question would be given an understanding of small oscillations (this is a problem from a PSET on small oscillations). Any help would be greatly appreciated!