Hamiltonian

The Hamiltonian is a concept used in classical and quantum mechanics as well as in related areas of mathematics, named after William Rowan Hamilton. In classical mechanics, the Hamiltonian H is a function on phase space, H(q,p,t), often interpreted as the total energy of the system expressed in generalized coordinates q and their conjugate momenta p. It is related to the Lagrangian L by a Legendre transform: p_i = ∂L/∂q̇_i and H(q,p,t) = ∑ p_i q̇_i − L(q,q̇,t). The equations of motion, known as Hamilton's equations, are dq_i/dt = ∂H/∂p_i and dp_i/dt = −∂H/∂q_i + ∂H/∂t. For time-independent Hamiltonians, H is conserved along trajectories. The formalism emphasizes phase-space structure and canonical transformations, which preserve the symplectic form and Poisson brackets.

In quantum mechanics, the Hamiltonian becomes an operator Ĥ acting on a Hilbert space. It generates time evolution through the Schrödinger equation iħ ∂ψ/∂t = Ĥ ψ. The energy spectrum is the set of eigenvalues of Ĥ for time-independent cases, and stationary states have time dependence e^−iEt/ħ.

In mathematics and other contexts, the term also appears in graph theory as Hamiltonian path or Hamiltonian cycle—paths or cycles that visit each vertex exactly once—named after the 19th-century mathematician William Rowan Hamilton; this use is distinct from the dynamical Hamiltonian concepts.