I guess the important part of the hamiltonian construction may be just having the next state depend on x(t) and x(t-1) (apparently those are called second-order cellular automata). Once you do that it’s relatively easy to make them reversible, you just need the dependence of x(t+1) on x(t-1) to be a permutation. But I don’t know whether using finite differences for the hamiltonian will easily give you conservation of momentum + energy in the same way that it would with derivatives.
I guess the important part of the hamiltonian construction may be just having the next state depend on x(t) and x(t-1) (apparently those are called second-order cellular automata). Once you do that it’s relatively easy to make them reversible, you just need the dependence of x(t+1) on x(t-1) to be a permutation. But I don’t know whether using finite differences for the hamiltonian will easily give you conservation of momentum + energy in the same way that it would with derivatives.