HamiltonFunktion

HamiltonFunktion, or Hamiltonian function, is a central object in Hamiltonian mechanics. It is a function H(p, q, t) defined on phase space that describes a system’s total energy and generates its time evolution. The formalism is named after William Rowan Hamilton, who developed the approach in the 19th century.

Definition and construction: For a system with generalized coordinates q_i and conjugate momenta p_i, defined by p_i = ∂L/∂q̇_i from a Lagrangian L(q, q̇, t), the Hamiltonian is H(q, p, t) = ∑ p_i q̇_i − L(q, q̇, t). If one can invert p_i = p_i(q, q̇, t) to obtain q̇_i = q̇_i(q, p, t), substituting yields H as a function of q, p, and t. When L has no explicit time dependence, H equals the total energy T + V.

Equations of motion: The time evolution is given by Hamilton’s equations: q̇_i = ∂H/∂p_i and ṗ_i = −∂H/∂q_i + ∂H/∂t. The Hamiltonian generates a flow on phase space and preserves the underlying symplectic structure. If H is time-independent, energy is conserved.

Relation to quantum mechanics: The Hamiltonian operator Ĥ is the quantization of H and governs dynamics through the Schrödinger equation iħ ∂ψ/∂t = Ĥ ψ. In the semiclassical limit, classical trajectories follow from Hamilton’s equations.

Examples and applications: For a single particle of mass m in a potential V(q, t), H = p^2/(2m) + V(q, t). In many-body systems, H includes kinetic terms, interparticle interactions, and external fields. The Hamiltonian formalism is foundational in classical dynamics, quantum theory, statistical mechanics, and computational physics.