To best help people there are very rare instances where its important to know why someone wants to know the answer to their question. This is such an instance. Why do you ask?
Regarding the question on the Hamiltonian its not clear what context you're speaking about. I suspect it might be in regards to quantum mechanics. Is that correct? In either case one doesn't "solve" a Hamiltonian anymore than one can solve the expression f(x) x^2.
In classical mechanics the Hamiltonian is an function and in quantum mechanics its an operator which is obtained from the classical Hamiltonian. The classical Hamiltonian is defined as the summation the product of the generalized coordinates and the corresponding generalized momentum minus the Lagrangian. Its defined per a given system.
There is no general way to solve the Schrodinger equation which is a differential equation. There are many methods to solve such equations and its different for each system.