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:…
Hamilton's principle (action)
Hamilton 1834: physical trajectory extremises action S = ∫L dt where L is Lagrangian. Euler-Lagrange equations follow. Backbone of…
Legendre condition (second variation)
Necessary condition for minimum: ∂²L/∂(ẏ)² ≥ 0 along extremal path. Strong + weak Legendre conditions. Conjugate-point analysis (Jacobi)…
Brachistochrone problem (Bernoulli)
Bernoulli 1696 challenge: curve of fastest descent under gravity is cycloid x = R(θ - sin θ), y = R(1 - cos θ). Solved by Newton, Leibniz,…
Plateau problem (minimal surfaces)
Plateau 1849 / Douglas-Rado 1931: every Jordan curve in ℝ³ bounds a minimal surface (zero mean-curvature). Soap-film physical realisation.…
Direct method (Tonelli)
Tonelli 1915: lower-semicontinuous coercive functional on weakly-closed sequentially-compact set attains minimum. Modern existence-proofs…
Noether current & conservation laws (extended)
Noether 1918 (extended): every continuous symmetry of action yields conserved current j^μ via ∂_μ j^μ = 0. Translation→energy-momentum;…
Euler-Lagrange equation
Euler 1744 + Lagrange 1755: stationary points of integral functional J[y] = integral L(x, y, y') dx satisfy d/dx(dL/dy') - dL/dy = 0;…
Noether's theorem (1918)
E Noether 1918: continuous symmetry of action ⇒ conserved current; translation→momentum, rotation→ang-mom, time→energy, gauge→charge;…
Hamilton's principle (least action)
Hamilton 1834: physical trajectory extremizes action S = integral L dt; Maupertuis-Euler abbreviated action W; Feynman 1948 path-integral…
Direct method (Tonelli-Weierstrass)
Tonelli 1915 direct method: existence of minimizer via lower-semicontinuity + coercivity in suitable function space (Sobolev W^{1,p});…
Isoperimetric problem (Dido)
Dido of Carthage problem (~800 BCE): max area enclosed by perimeter L is L^2/(4 pi) (circle); generalized via Lagrange-multipliers to…
Pontryagin maximum principle (1956)
Pontryagin-Boltyanskii-Gamkrelidze-Mishchenko 1956 maximum principle: optimal control u*(t) maximizes Hamiltonian H(x, p, u); foundation of…
Brachistochrone (Bernoulli 1696)
Johann Bernoulli 1696 brachistochrone problem; modern direct + indirect methods + Pontryagin maximum.
Dirichlet principle (1850)
P Dirichlet 1850s + Hilbert 1900 minimization of Dirichlet integral; foundation of PDE existence theory.
Quasi-convexity (Morrey 1948)
C Morrey 1948 quasi-convexity weak-lower-semicontinuity; modern direct-method + nonlinear-elasticity + image-analysis.
Optimal transport (Monge 1781 / Kantorovich 1942)
Monge 1781 + Kantorovich 1942 (Nobel 1975) optimal-transport; modern Brenier + Villani Fields 2010 + ML.
HJB (Bellman 1957)
R Bellman 1957 dynamic-programming; modern Hamilton-Jacobi-Bellman PDE + viscosity-solutions Crandall-Lions 1983.
Gamma-convergence (De Giorgi 1975)
E De Giorgi 1975 Gamma-convergence; modern Modica-Mortola + Allen-Cahn + sharp-interface phase-field.