Natural frequency of 3 coulomb force bound particles in EF

[Franky4]

Homework Statement

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.

Homework 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.

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

 

[mfb]

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

Advanced is fine.

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.

 

[Franky4]

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

 

[Franky4]

Advanced is fine.

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.

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 ?

 

[mfb]

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