Natural frequency of 3 coulomb force bound particles in EF

1. Mar 1, 2015

Franky4

1. The problem statement, all variables and given/known data
I was given a task to model (using Matlab) 3 identical particles in external field and find spectra of lowest system energy states using gradient descend method for each particle in the system.
I did a run of 500 random generated coordinates and found this distribution.
(blue dots are final coordinates, red are initial)
Now I need to find natural frequency and vectors for the lowest energy state system of 4.5299 units in this case. Also find eigen vectors for different nodes.

2. Relevant equations
Potential due to external field: U(x,y) = x^3 + x^4 - 0.1*x + y^6 - x*y;
Potential due to interactions between particles: U(r1, r2) = a/(abs(r1 - r2)); a is a constant.

3. The attempt at a solution
I found plenty of information for systems with masses, springs and stiffness coefficients, but nothing for this and I am not sure which way to proceed. Should I be trying to think of this three particle bound system as three masses (charges in this case) connected with different springs (coulomb force) for which stiffness k should be calculated from the force by the gradient of external field and repulsion force between particles or am I overthinking this.

First time posting, not sure between introductory and advanced homework sections

2. Mar 2, 2015

Staff: Mentor

You can calculate the energy of the system (relative to the ground state) as function of the positions of the objects relative to the ground state (assuming small deviations). That gives a 6-dimensional phase space where you can find eigenvectors and eigenvalues corresponding to different oscillations.

3. Mar 2, 2015

Franky4

Thanks. I had a talk with my teacher and told me to read up about Hessian matrix.

4. Mar 4, 2015

Franky4

I calculated eigenvalues and eigenvectors of the Hessian matrix where F(x1,x2,x3,y1,y2,y3) { (xi, yi) being a pair of coordinates of the partcile} is a total potential energy of the system. I am asked to create a visual of different mode eigenvectors, but they are 6 dimensional, how am I to do this ?

5. Mar 4, 2015

Staff: Mentor

You can draw them as displacement directions for the three particles - three arrows for each mode.