The Hamiltonian is a quantity of great importance in both classical and quantum mechanics.
In classical dynamics, the Hamiltonian is defined to be
For a mass attached to a spring of spring constant extended by a distance , and so
which is the familiar expression for the energy of a simple harmonic oscillator.
The Hamiltonian for many quantum mechanical systems can be obtained by writing down a corresponding classical Hamiltonian and promoting all of the co-ordinates and momenta to operators. The quantum mechanical Hamiltonian is of central importance to the Schrodinger equation.