Hamiltoniano

Hamiltoniano, or Hamiltonian in English, is a term used in physics and mathematics for quantities associated with Hamilton's formulation of dynamics. Named after William Rowan Hamilton, the concept appears in classical mechanics, quantum mechanics, and related fields. In languages such as Spanish, Portuguese, or Italian, Hamiltoniano is the common term.

In classical mechanics, the Hamiltonian H(q,p,t) is a function on phase space that often represents the total energy, expressed in terms of generalized coordinates q and conjugate momenta p. Hamilton's equations are dq_i/dt = ∂H/∂p_i and dp_i/dt = -∂H/∂q_i. If H is time-independent, energy is conserved. The formalism emphasizes phase space and canonical coordinates.

In quantum mechanics, the Hamiltonian operator Ĥ governs time evolution via iħ ∂ψ/∂t = Ĥ ψ and corresponds to the total energy observable. Its spectrum determines energy levels and dynamics. In statistical mechanics, exp(-βĤ) defines equilibrium ensembles. The Hamiltonian formalism also extends to quantum field theory and many-body physics through a Hamiltonian density.

Relation to the Lagrangian framework is given by the Legendre transform, converting L(q, q̇, t) to H(q,p,t). The approach yields canonical transformations, Poisson brackets, and, in quantum theory, path integral formulations. The term Hamiltoniano is widely used in physics and is essential for modeling energy and evolution in a wide range of systems.