§ DICTIONARY · CONCEPT
Hamiltonian
A scalar function H(q, p, t) whose partial derivatives, via Hamilton's equations, generate time evolution. For conservative systems, H = T + V.
§ 01
Definition
The Hamiltonian is the scalar function H(q, p, t) obtained from the Lagrangian by a Legendre transform: H = Σ pᵢq̇ᵢ − L, with pᵢ = ∂L/∂q̇ᵢ. For a time-independent potential it equals the total energy T + V — a consequence of time-translation symmetry via Noether's theorem.
The Hamiltonian generates the dynamics through Hamilton's equations: q̇ᵢ = ∂H/∂pᵢ, ṗᵢ = −∂H/∂qᵢ. These are first-order and symmetric in q and p, which makes them the natural setting for phase-space geometry, symplectic integrators, and the canonical quantisation of classical systems into quantum mechanics.