Analogous to what the Euler-Lagrange equation is to Lagrangian mechanics, Hamilton's equations give the equations of motion from a given input Hamiltonian:
So once you have the Hamiltonian, you can write down this system of partial differential equations which can then be numerically solved.

$∂q_{j}∂H =−p˙ _{j}∂p_{j}∂H =q˙ _{j}$