A particle of mass m described by one generalized coordinate q moves under the influence of a potential V(q) and a damping force −2mγq˙  proportional to its velocity with the Lagrangian  L = e2γt(1/2 * mq˙2 − V (q)) which gives the desired equation of motion. (a) Consider the following generating function:  F = eγtqP - QP. Obtain the canonical transformation from (q,p) to (Q,P) and the transformed Hamiltonian K(Q,P,t). (b) Let V (q) = (1/2)mω2q2 be a harmonic potential with a natural frequency  ω and note that the transformed Hamiltonian yields a constant of motion.  Obtain the solution Q(t) for the damped oscillator in the under damped case γ < ω by solving Hamilton's equations in the transformed coordinates. Then, write down the solution q(t) using the canonical coordinates obtained in part (a).

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