Like the Newtonian and Lagrangian techniques, the Hamiltonian is yet another completely equivalent way to determine system dynamics. In most cases, the Hamiltonian is defined as the sum of kinetic and potential energy
The paper that the group wrote concerns chaotic systems, where small changes to conditions can cause wildly different results. Often a chaotic system will have a subset of simple results – I actually wrote about an example long ago, but since I hadn't yet taken any nonlinear dynamics, I wasn't sure if it qualified. If you're trying to train a neural network to predict a system's dynamics, this transition between simple and chaotic motion can cause problems.
One of the example systems the paper studies is a circular billiards table, with a peg in the center:
The group's idea was to change the way the neural network is trained, incorporating the Hamiltonian to give it awareness of the chaotic nature of the system. They compared the results from a standard network, a Hamiltonian network, and from solving the differential equations: