Question

A particle of mass m moves under the influence of gravity along the helix z =kθ, r =constant, where k is a constant and z is vertical. Obtain the Hamiltonian equations of motion.

Answer 1

$z=k\theta,$

$L=T-U,$

$T=\frac{km\theta^2}2,$

$U=kmg\ddot{\theta},$

$\frac{\partial L}{\partial \theta}-\frac d{dt}\frac{\partial L}{\partial \dot {\theta}}=0,$

$km\theta\dot{\theta}-mg\ddot{\theta}=0.$