Ex 1.22. Driven pendulum

Structure and Interpretation of Classical Mechanics

Show that the Lagrangian (1.89) can be used to describe the driven pendulum, where the position of the pivot is a specified function of time: Derive the equations of motion using the Newtonian constraint force prescription, …

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[guess]

$\displaystyle{ \begin{aligned} m \ddot{y} &= F \cos \theta - mg \\ m \ddot{x} &= - F \sin \theta \\ \end{aligned}}$

$\displaystyle{ \begin{aligned} m \ddot{y} &= F \frac{y_s - y}{l} - mg \\ m \ddot{x} &= - F \frac{x - x_s}{l} \\ \sqrt{(x_s - x)^2 + (y_s - y)^2} &= l \\ \end{aligned}}$