Derive the equations of motion using Lagrange's equations
The Lagrange's equation is given by:
\(\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{\theta_i}}\right)-\frac{\partial L}{\partial \theta_i}=0\)
Where \(i\) represents the angle \(\theta_1\), \(\theta_2\), and \(\theta_3\) for each pendulum bob.
Applying Lagrange's equations for each angle and simplifying the expressions, we obtain the following equations of motion:
\(\frac{d^2\theta_1}{dt^2} = f_1(\theta_1, \theta_2, \theta_3, \dot{\theta}_1, \dot{\theta}_2, \dot{\theta}_3, F_1, F_2, F_3)\)
\(\frac{d^2\theta_2}{dt^2} = f_2(\theta_1, \theta_2, \theta_3, \dot{\theta}_1, \dot{\theta}_2, \dot{\theta}_3, F_1, F_2, F_3)\)
\(\frac{d^2\theta_3}{dt^2} = f_3(\theta_1, \theta_2, \theta_3, \dot{\theta}_1, \dot{\theta}_2, \dot{\theta}_3, F_1, F_2, F_3)\)
These equations represent the equations of motion for the triple pendulum subjected to horizontal forces \(F_1, F_2,\) and \(F_3\).