Calc. Variations·Course
Calculus of Variations
Calculus of variations: functionals, Euler-Lagrange equation, second-order conditions, and links to mechanics and optimal transport
Part of the Mathematics track — finish it for a verifiable diploma →
§ 01 — Orientation
New here?
What you'll be able to do, who this is for, how long it takes, and where to begin.
By the end, you will be able to
- 01Set up and solve variational problems.
- 02Derive Euler–Lagrange equations.
- 03Check second-order conditions.
- 04Use the Hamiltonian formalism.
- 05Apply variational methods in physics and economics.
Who this is for
How long it takes
- Quick orientationSkim the opening module and the cheatsheet to get the shape of it.~1 h
- Full read-throughRead every article once, in order.~1 h
- Mastery pathRead, take the quizzes, and space out your reviews.~2 weeks
- APA
Stoa. (2026). Calculus of Variations [Online course]. Stoa. https://stoa.school/course/calculus-variations
- MLA
Stoa. “Calculus of Variations.” Stoa, 2026, https://stoa.school/course/calculus-variations.
- Chicago
Stoa. “Calculus of Variations.” Stoa. Accessed September 1, 2026. https://stoa.school/course/calculus-variations.
§ 02 — Curriculum
4 modules.
Each module is a small unit. Most read in sequence — but a determined reader can begin anywhere.
- M IFoundations of the Calculus of VariationsFunctionals, the first variation, and the Euler–Lagrange equation3 articles
18 minBegin → - M IIThe Bolza Problem and Boundary ConditionsGeneralized formulations of the calculus of variations and transversality conditions3 articles
18 minBegin → - M IIIHamilton–Jacobi Theory and Geometrical OpticsThe Hamilton–Jacobi equation, optimal transport, and geometric applications3 articles
18 minBegin → - M IVModern Applications of the Calculus of VariationsShape optimization, continuum mechanics, and Noether’s theorem3 articles
18 minBegin →
§ 03 — Learning outcomes
4 outcomes.
Compute the first variation of a functional and derive the Euler–Lagrange equation
Study the second variation and the Legendre and Jacobi conditions
Apply the Legendre transform and the Hamilton–Jacobi equation.
Solve problems involving geodesics, minimal surfaces, and optimal transport.
§ 04 — Practices