1. (20 points) Verify the Jacobi identity for the Poisson brackets.
2. (20 points) Show by the use of Poisson brackets that for a one-dimensional harmonic
oscillator there is a constant of motion u defined as
u(q, p, t) = ln(p + imωq) − iωt, ω =
√
k
m
. (1)
What is the physical significance of it?
3. (35 points) Show that the following transformation is canonical (α is a fixed parameter):
x =
1
α
(
√
2P1 sin Q1 + P2), px =
α
2
(
√
2P1 cos Q1 − Q2),
y =
1
α
(
√
2P1 cos Q1 + Q2), py = −
α
2
(
√
2P1 sin Q1 − P2), (2)
Apply this transformation to the problem of a particle of charge q moving in a plane that is
perpendicular to a constant magnetic field B⃗ . Express the Hamiltonian for this problem in the
(Qi
, Pi) coordinates letting the parameter α take the form
α
2 =
qB
c
. (3)
From this Hamiltonian, obtain the motion of the particle as a function of time.
4. (25 points) Use the method of infinitesimal canonical transformations to solve the motion
of a one-dimensional harmonic oscillator as a function of time.
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