Chapter 7, Problem 7.32P

A particle of mass m described by one generalized coordinate q movesunder 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).

Prove the following: if the Hamiltonian is independent of time, then ∆E doesn't change in time. Show work and be explicit to prove the statement.

For a one dimensional system, x is the position operator and p the momentum operator in the x direction.Show that the commutator [x, p] = ih