(a) Let H(u, v) = au2 + buv + cv2 be a quadratic function. Prove that the non-equilibrium trajectories of the associated Hamiltonian system and those of the gradient flow are mutually orthogonal, i.e.. they always intersect at right angles.
(b) Verify this result for
(i) u2 + 3v2
(ii) uv
by drawing representative trajectories of both systems on the same graph.