Ok, so I read the article, and 1) they did not use Hamiltonian mechanics whatsoever in their analysis, and 2) the way they modeled the cannon motion is very, very flawed. They've modeled the cannon position as proportional to -phi/phi_max where phi is the tilt angle of the ship. According to this model, as soon as the ship begins rocking back from its maximum angular displacement, the cannonballs will also change directions and begin rolling in the opposite direction, as evidenced by figure 7 in the article. This is not physical whatsoever. Consider a jar containing a marble. Tilting the jar left will cause the marble in the jar to shift to the left side of the jar. Then slowly tilt the jar back to the equilibrium position (completely upright), and then tilt the jar to the right. Notice how the marble will stay at the leftmost part of the jar until after the jar is tilted past the equilibrium point and then will begin rolling to the right side of the jar, not immediately when you begin to tilt the jar back to the equilibrium position from the leftmost tilt. Modeling the cannonballs correctly i.e. having them move this way will introduce a damping factor in their model and will make it harder for the people running back and forth on the ship to overturn it.