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 O P 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 m L, find the tension in the string at time t.