Paper 2, Section I, A

Classical Dynamics | Part II, 2012

(a) The action for a system with a generalized coordinate qq is given by

S=t1t2L(q,q˙,t)dtS=\int_{t_{1}}^{t_{2}} L(q, \dot{q}, t) d t

State the Principle of Least Action and state the Euler-Lagrange equation.

(b) Consider a light rigid circular wire of radius aa and centre OO. The wire lies in a vertical plane, which rotates about the vertical axis through OO. At time tt the plane containing the wire makes an angle ϕ(t)\phi(t) with a fixed vertical plane. A bead of mass mm is threaded onto the wire. The bead slides without friction along the wire, and its location is denoted by AA. The angle between the line OAO A and the downward vertical is θ(t)\theta(t).

Show that the Lagrangian of this system is

ma22θ˙2+ma22ϕ˙2sin2θ+mgacosθ.\frac{m a^{2}}{2} \dot{\theta}^{2}+\frac{m a^{2}}{2} \dot{\phi}^{2} \sin ^{2} \theta+m g a \cos \theta .

Calculate two independent constants of the motion, and explain their physical significance.

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