Find the Lagrangian of simple pendulum?

Answer 1

The generalized coordinate for the pendulum is the angle of the arm off vertical, theta. Theta is 0 when the pendulum arm is down and pi when the arm is up.

M = mass of pendulum

L = length of pendulum arm

g = acceleration of gravity

\theta = angle of pendulum arm off vertical

\dot{\theta} = time derivative of \theta

What are the kinetic and potential energies?

Kinetic energy: T = (1/2)*M*(L*\dot{\theta})^2

Potential energy: V' = MLg(1-cos(\theta))

V = -MLg*cos(\theta)

--note: we can shift the potential by any constant, so lets choose to drop the MLg

The Lagrangian is L=T-V: L = (1/2)ML^2\dot{\theta}^2 + MLg*cos(\theta)