What is the Hamiltonian equation

For starter what is Hamiltonian, the Hamiltonian of a system is defined to be the sum of the kinetic and potential energies expressed as a function of positions and their conjugate momenta.

The Hamiltonian formulation of mechanics basicaly describes a system in terms of generalised coordinates  and generalised momenta .

Explanation:

Hamiltonian equations: The solution of Hamilton's equations of motion will yield a trajectory in terms of positions and momenta as functions of time.

Hope this helped...I tried to make it simple as I could. :)