Physics - Advanced Classical Mechanics - Hamiltonian mechanics - Hamiltonian Formulation

4-step lesson. Step 1: 'The Hamiltonian: Starting from L(q, q̇, t), define canonical momentum p = ∂L/∂q̇. Perform Legendre transform: H(q, p, t) = pq̇ - L (expressed in terms of q and p, not q̇). For many systems: H = T + V = total energy. Hamilton's equations: q̇ᵢ = ∂H/∂pᵢ, ṗᵢ = -∂H/∂qᵢ. Beautiful