Text: Need help with b and c, please explain steps.
1. Consider the following Hamiltonian
H = +Vq^2
V(q) = |q|.
a) Derive Hamilton's equations. Is the Hamiltonian vector field continuous?
b) Solve Hamilton's equations for every real time assuming that the energy H remains the same on a solution while passing from positive to negative q. Use the initial conditions q0 = H and p(0) = 0 for a solution (q(t), p(t)). By using the solution to Hamilton's equation, show that the motion is periodic of period T = 4/2H.
c) Show that with H > 0, the solution for the coordinate q goes to q(t) = 0 indicating that one can physically extend the Hamiltonian vector field to include a fixed point (qt, pt) = (0, 0). Can you identify which type of fixed point is this even though V(q) is not a priori defined in q = 0?
d) Sketch a graph of (i) the potential V(q) and (ii) the phase portrait of the system (you might do it both by hand or with a computer). Make sure that you identify the type of fixed points, and sketch the direction of the vector field flow on orbits.
e) Given the motion of the particle starts with a positive momentum at q = 0, show that the angle variable for the motion as a function of position, q, for the first quarter period is 9(q) = 1-.