In physics, Hamiltonian has distinct but closely related meanings.

• In classical mechanics, the Hamiltonian is a function describing the state of a mechanical system in terms of position and momentum variables. See Hamiltonian mechanics.
• In quantum mechanics, the Hamiltonian is an operator corresponding to the total energy of a system. See Hamiltonian (quantum mechanics).

In graph theory, a graph is Hamiltonian if it contains a path that starts and ends at the same vertex and includes each vertex exactly once. Such a path is called a Hamiltonian cycle.

In group theory, see Hamiltonian group.

Both the Hamiltonian operator in physics and Hamiltonian cycles in graph theory are named after Sir William Rowan Hamilton.