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
. (1)
What is the physical significance of it?
3. (35 points) Show that the following transformation is canonical (α is a fixed parameter):
, (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
. (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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