Optimisation of functionals — maps from spaces of functions to the reals — via stationary-action principles. Classical Johann Bernoulli 1696 brachistochrone problem launched the field; Euler 1744 and Lagrange 1755-60 derived the…
calculus-of-variations
Euler-Lagrange equation d/dt(∂L/∂q̇) − ∂L/∂q = 0
Euler 1744 / Lagrange 1755-60 — a functional S[q] = ∫_{t_1}^{t_2} L(q, q̇, t) dt is stationary under endpoint-fixed variations δq iff…
Noether 1918: continuous symmetry ⇒ conserved current
Emmy Noether 1918 (Gött. Nachr. 235) — for every continuous one-parameter symmetry of the action (q → q + ε·X(q) with δS = ε·dK/dt) there…
Isoperimetric inequality: 4πA ≤ P² with equality iff circle
For every simple closed plane curve of perimeter P enclosing area A, the isoperimetric inequality 4πA ≤ P² holds with equality iff the…
EL on L = mv²/2 − kx²/2: equation k·x + m·ẍ = 0, residual ≡ 0 on A·cos(ωt+φ)
Exact symbolic application of the Euler-Lagrange equation to the harmonic-oscillator Lagrangian L = m·ẋ²/2 − k·x²/2. Computation: ∂L/∂ẋ =…
Noether energy conservation: H = kx²/2 + mẋ²/2 = const, dH/dt ≡ 0
Explicit Noether-theorem conservation of energy for the harmonic-oscillator Lagrangian. The Lagrangian L = m·ẋ²/2 − k·x²/2 has no explicit…
Isoperimetric ratio: 4πA/P² = 1 for circle (exact)
Exact symbolic confirmation that the circle saturates the isoperimetric inequality with ratio exactly 1. For a circle of radius r:…