Hamiltonian Formalism
A formulation of classical mechanics describing system dynamics in 2N-dimensional phase space using position q_i and canonical momentum p_i.
Governing formula
H(q, p, t) = ∑ p_i q̇_i - L, q̇_i = ∂H / ∂p_i, ṗ_i = -∂H / ∂q_i
SI unit
Joules (Energy)
In depth
Formulated by William Rowan Hamilton in 1833. Converts N second-order Euler-Lagrange equations into 2N first-order canonical differential equations, forming the direct bridge to quantum mechanics.
Examples in the real world
Mapping phase space trajectories and formulating quantum Hamiltonian operators Ĥ in quantum mechanics.