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) Obtain the Hamiltonian H(q,p,t) for this system. (b) 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). (c) Pick V (q) = (1/2)mω2q2 a as a harmonic potential with a natural frequency ω. Show that the transformed Hamiltonian yields a constant of motion.
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) Obtain the Hamiltonian H(q,p,t) for this system. (b) 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). (c) Pick V (q) = (1/2)mω2q2 a as a harmonic potential with a natural frequency ω. Show that the transformed Hamiltonian yields a constant of motion.
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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) Obtain the Hamiltonian H(q,p,t) for this system.
(b) 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).
(c) Pick V (q) = (1/2)mω2q2 a as a harmonic potential with a natural frequency ω. Show that the transformed Hamiltonian yields a constant of motion.
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