Calc. Variations·Course
Calculus of Variations
Calculus of variations: functionals, Euler-Lagrange equation, second-order conditions, and links to mechanics and optimal transport
4
Modules
12
Articles
~1 h
Reading
IV
CLOs
§ 01 — 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 →
§ 02 — Learning outcomes
4 outcomes.
CLO I
Functionals and the Euler–Lagrange Equation
Compute the first variation of a functional and derive the Euler–Lagrange equation
CLO II
Second-Order Conditions
Study the second variation and the Legendre and Jacobi conditions
CLO III
Canonical Formalism
Apply the Legendre transform and the Hamilton–Jacobi equation.
CLO IV
Applications
Solve problems involving geodesics, minimal surfaces, and optimal transport.
§ 03 — Practices