In classical dynamics, the Hamiltonian is defined to be
The Hamilton equations are:
which is the familiar expression for the energy of a simple harmonic oscillator.
The equations of motion are:
Inserting this into the first equation, we get . This is just Newton's second law, F = ma.
The Hamiltonian for many quantum mechanical systems can be obtained by writing down a corresponding classical Hamiltonian and promoting all of the coordinates and momenta to operators. The quantum mechanical Hamiltonian is of central importance to the Schrodinger equation.