Stability Analysis of Equilibrium Solutions using Small Perturbations

[curiousPep]

Homework Statement

How can I find the condition for the system to be equilibrium and how to check if these solution are stable or not?

Relevant Equations

Equations of motion

When I use Lagrange to get the equations of motion, in order to find the equilibrium conditions I set the parameters q as constants thus the derivatives to be zero and then calculate the q's that satisfy the equations of motion obtained.
In ordert to check about stability I think I need to add some small perturbation, thus q' = q+dq. Then how can I check if the solutions are stable or not?

 

[ergospherical]

There are various possible approaches, i.e. you may check whether the small perturbation varies simple harmonically around the equilibrium position (that is, put $q = q_0 + \epsilon$ into the E.O.M. and try to arrive at an equation of the form $\ddot{\epsilon} + \omega^2 \epsilon = 0$).