{
  "id": "calculus-of-variations",
  "domain": "applied",
  "type": "Concept",
  "title": "Calculus of Variations",
  "slug": "calculus-of-variations",
  "url": "/mathematics/applied/calculus-of-variations/",
  "summary": "Optimises functionals (integrals depending on unknown functions). Brachistochrone 1696 (Bernoulli's challenge). Euler-Lagrange equation 1744/1755. Plateau's problem (Douglas-Radó 1930-31, Fields Medal 1936). Optimal control theory (Pontryagin 1950s-60s).",
  "added_to_codex": "2026-07-09",
  "last_verified": "2026-07-09",
  "sources": [
    {
      "tier": 1,
      "citation": "Euler, L. (1744). Methodus inveniendi lineas curvas. Lausanne."
    },
    {
      "tier": 1,
      "citation": "Gelfand, I.M. and Fomin, S.V. (2000). Calculus of Variations. Dover."
    },
    {
      "tier": 2,
      "citation": "Pontryagin, L.S. et al. (1962). The Mathematical Theory of Optimal Processes."
    },
    {
      "tier": 3,
      "citation": "MacCluer, C.R. (2004). Calculus of Variations. Pearson."
    }
  ],
  "entry_status": "live",
  "reading_levels_available": [
    "curious",
    "exploring",
    "deep_dive"
  ],
  "completeness": 1.0,
  "codex_version": "2.5"
}