Least Action Principle / Hamilton's Principle
Overview
The actual path of a system extremizes the action S. For mechanics, S = ∫ L dt with L = T − V. Yields Euler–Lagrange equations; underlies classical mechanics, field theory, GR.
Era: 1744–1834
Proposed at: Basel / Berlin / Turin / Dublin
Formal structure
Domain: Paths γ: [t₁, t₂] → M (configuration space)
S[γ] = ∫_{t₁}^{t₂} L(q, q̇, t) dt
Constraints: δγ(t₁) = δγ(t₂) = 0 (fixed endpoints)
Objective: δS = 0 ⇒ ∂L/∂q − (d/dt)(∂L/∂q̇) = 0 (Euler–Lagrange)