5. Consider the case of a free particle moving in the plane under the influence of an
r = r(t) is the radius of the particle's position and p = p(t) is the corresponding radial momentum. Here k > 0, as arises for an attracting gravitational force. First, show that H is conserved along trajectories of the Hamiltonian ODE
He Op
OH ar
p=
plot several representative level sets of H in the (r,p) plane, showing different trajectories of the ODE system, making sure to plot level sets along which H is negative, zero, and positive; also, briefly describe the motion of the particle in the effective potential for each of these three cases.