A II-1

Non-Relativistic

Hamiltonian Trajectories

AII-2

Relativistic

Hamiltonian Trajectories

${\begin{cases} \frac{d r}{d t} = \frac{\partial H}{\partial p} \equiv \frac{p}{m} \\ \frac{d p}{d t} = - \frac{\partial H}{\partial r} \equiv - \nabla [V (r) + W (r, E)] \\ \nabla \cdot (R^{2} p) = 0 \\ | p (t = 0) | \equiv p_{0} = \sqrt{2 m E} \end{cases}$

${\begin{cases} \frac{d r}{d t} = \frac{\partial H}{\partial p} \equiv \frac{c^{2} p}{E - V (r)} \\ \frac{d p}{d t} = - \frac{\partial H}{\partial r} \equiv - \nabla V (r) - \frac{E}{E - V (r)} \nabla W (r, E) \\ \nabla \cdot (R^{2} p) = 0 \\ | p (t = 0) | \equiv p_{0} = \sqrt{{(E / c)}^{2} - {(m_{0} c)}^{2}} \end{cases}$

$W (r, E) = - \frac{ℏ^{2}}{2 m} \frac{\nabla^{2} R (r, E)}{R (r, E)}$

$W (r, E) = - \frac{ℏ^{2} c^{2}}{2 E} \frac{\nabla^{2} R (r, E)}{R (r, E)}$