Paper 2, Section I, A
(a) The action for a system with a generalized coordinate is given by
State the Principle of Least Action and state the Euler-Lagrange equation.
(b) Consider a light rigid circular wire of radius and centre . The wire lies in a vertical plane, which rotates about the vertical axis through . At time the plane containing the wire makes an angle with a fixed vertical plane. A bead of mass is threaded onto the wire. The bead slides without friction along the wire, and its location is denoted by . The angle between the line and the downward vertical is .
Show that the Lagrangian of this system is
Calculate two independent constants of the motion, and explain their physical significance.
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