Source: cirosantilli/hamilton-s-equations
= Hamilton's equations
{c}
{wiki}
Analogous to what the <Euler-Lagrange equation> is to <Lagrangian mechanics>, Hamilton's equations give the <equations of motion> from a given input <Hamiltonian>:
$$
\frac{\partial H}{\partial q_j} = -\dot{p}_j \\
\frac{\partial H}{\partial p_j} = \dot{q}_j
$$
So once you have the Hamiltonian, you can write down this system of <partial differential equations> which can then be numerically solved.