# Calculus of Variations

**Type:** Concept  
**Domain:** applied  
**Codex URL:** /mathematics/applied/calculus-of-variations/  
**Entry status:** Live — v1.0 (2026-07-09)

## 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).

## Sources
- [Tier 1] Euler, L. (1744). Methodus inveniendi lineas curvas. Lausanne.
- [Tier 1] Gelfand, I.M. and Fomin, S.V. (2000). Calculus of Variations. Dover.
- [Tier 2] Pontryagin, L.S. et al. (1962). The Mathematical Theory of Optimal Processes.
- [Tier 3] MacCluer, C.R. (2004). Calculus of Variations. Pearson.

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*Mathematics Codex entry v1.0 — added 2026-07-09 — thecodex.expert/mathematics/*