Particle in Magnetic Field

a) Give a relationship between Hamilton’s equations under a canonical transformation. Verify that the transformation

is canonical.

b) Find Hamilton’s equations of motion for a particle moving in a plane in a magnetic field described by the vector potential

in terms of the new variables Q₁, Q₂, P₁, P₂ introduced above, using ω = eH/mc.

SOLUTION

a) A canonical transformation preserves the form of Hamilton’s equations:

where H = H(Q, P) is the transformed Hamiltonian. It can be shown that Poisson brackets are invariant under such a transformation. In other words, for two functions f, g

(1)

where q, p and Q, P are the old and new variables, respectively. Since we have the following equations for Poisson brackets:

(2)

(1) and (2) combined give equivalent conditions for a transformation to be canonical:

Let us check for our transformation (we let

and

Similarly

and so on.

For a particle in a magnetic field described by the vector potential A =(–YH/2,XH/2,0), which corresponds to a constant magnetic field we should use the generalized momentum P in the Hamiltonian

so the Hamiltonian

So the Hamiltonian H does not depend on Q₁, Q₂ and

where α is the initial phase. Also

Where X₀ and Y₀ are defined by the initial conditions. We can write this solution in terms of the variables X, Y, p_x, p_y:

Similarly

so this is indeed the solution for a particle moving in one plane in a constant magnetic field perpendicular to the plane.