Physics 3151A/B Lecture 6: Hamilton

A particle of mass m described by one generalized coordinate q moves under the influence of a potential V(q) and a damping force -2m gamma q proportional to its velocity. Show that the following Lagrangian gives the desired equation of motion. L = e2t(1/2 mq2 - V(q)) Obtain the Hamiltonian H(q.p.t) for this system. Consider the following generating function: F = eytqP-QP obtain the canonical transformation from (q.p) to (Q.P) and the transformed Hamiltonian K(Q,P,t). V(q) = 1/2 m omwga2 q2 Pick as a harmonic potential with a natural frequency. Show that the transformed Hamiltonian yields a constant of motion. Obtain the solution Q(t) for the damped oscillator in the under damped case gamma