Normal Mode Expansion of Damped Coupled Oscillators in 3 dimensions

In this paper, I aim to study free oscillations of a system of oscillators in more than one dimensions in the absence of damping. The basic approach lies in decoupling the motion in the individual perpendicular directions. Once the equations are decoupled, the existent techniques of Normal mode expansion for 1-dimensional oscillators are used to solve for the equations of motion. I also study the motion of a driven system of oscillators in higher dimensions in the presence of a velocity dependent damping force.