Engineering Stability Analysis of Equilibrium Solutions using Small Perturbations

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To analyze the stability of equilibrium solutions using small perturbations, the initial step involves setting parameters as constants in the equations of motion derived from Lagrange's method. This leads to the identification of equilibrium conditions by solving for the parameters that satisfy these equations. To assess stability, small perturbations are introduced, represented as q' = q + dq. The stability can be evaluated by substituting q = q_0 + ε into the equations of motion and determining if the resulting equation resembles the simple harmonic oscillator form, specifically ##\ddot{ε} + ω^2 ε = 0##. This approach allows for a clear assessment of whether the equilibrium solutions are stable.
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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?
 
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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##).
 
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