A pendulum consists of a bob on a light rigid rod of length
PHYSICS · Subjective · Fermi University Physics · Mechanics & Dynamical Systems · Oscillations and normal modes · Lagrangian mechanics
Action and Symmetry, Ch The action, the first variation and the boundary term
Problem
A pendulum consists of a bob on a light rigid rod of length ℓp=1.0m, free to swing in a vertical plane under gravity (g=9.8ms−2), released from rest at a small angle θ0 from the vertical. For small θ0 the restoring torque is proportional to θ to good approximation, so the swing is simple harmonic with angular frequency ω=g/ℓp. Up to a constant, its small-angle Lagrangian L=21mℓp2θ˙2−21mgℓpθ2 has the same quadratic form as that of the mass-spring oscillator, with mℓp2 in place of the mass, whose action was examined for a sinusoidal trial variation elsewhere. Estimate the time after release of the true swinging path's first conjugate point, using the oscillator result that this occurs at half a period, and say what that time means for whether the path minimises the action.