Question

4. Hamiltonian equations of motion A particle of mass m moving in two dimensions has the Hamiltonian H(x, y, px, py) = (px − cy) ^2 + (py + cx)^ 2 /2m where c is a constant. Find the equations of motion for x, y, px, py. Use these to write the equations of motion for the complex variables ζ = x + iy and ρ = px + ipy. Eliminate ρ to show that m¨ζ − 2ic ˙ζ = 0 . Find the general solution for ζ(t) and ρ(t). A particle has the initial position (x, y) = (0, 0) and initial momentum (px, py) = (mU, 0) at t = 0. Find its position, momentum and velocity as functions of t.

Answer 1

This is the correct sequence of answers.  

I have first computed the general equation of motion using the standard formula and then substituting these equations in other variables to compute Tao and rho and finally, I have computed position, momentum, and velocity.