A particle P is connected to one end of a light inextensible string which passes through a small hole O in a smooth horizontal table and extends below the table in a vertical straight line. P slides on the upper surface of the table while the string is pulled downwards from below in a prescribed manner. (Suppose that the length of the horizontal part of the string is R(t) at time t.) Take θ, the angle between OP and some fixed reference line in the table, as generalised coordinate and obtain Lagrange's equation. Show that θ is a cyclic coordinate and find (and identify) the corresponding conserved momentum pθ. Why is the kinetic energy not conserved?
If the constant value of pθ is mL, find the tension in the string at time t.