Pendulum with a shortening string A particle is suspended from a support by a light inextensible string which passes through a small fixed ring vertically below the support. The particle moves in a vertical plane with the string taut. At the same time, the support is made to move vertically having an upward displacement Z(t) at time t. The effect is that the particle oscillates like a simple pendulum whose string length at time t is a−Z(t), where a is a positive constant. Show that the Lagrangian is
L=21m((a−Z)2θ˙2+Z˙2)+mg(a−Z)cosθ,
where θ is the angle between the string and the downward vertical.
Find the Hamiltonian and obtain Hamilton's equations. Is H conserved?