Mathematics / advanced
Calculus of variations
Derive equations and stability from optimization over functions, while distinguishing stationarity from existence.
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Module 01 advanced
Calculus of variations
Derive equations and stability from optimization over functions, while distinguishing stationarity from existence.
- 01
Functionals and first variations
Differentiate an objective over admissible functions while keeping the function space, boundary values, and topology explicit.
28 min - 02
Euler–Lagrange equations and boundary terms
Derive an interior differential equation without losing the endpoint information carried by integration by parts.
29 min - 03
Natural boundary conditions and transversality
Derive endpoint conditions from free endpoint motions, terminal costs, and moving terminal curves.
30 min - 04
Constrained variations and multipliers
Restrict perturbations to a constraint tangent space and derive a valid multiplier equation under a constraint qualification.
30 min - 05
Noether identities for variational symmetries
Derive conserved quantities from smooth continuous symmetries and identify exactly where the Euler–Lagrange equation is used.
30 min - 06
Second variation and the Legendre condition
Compute the quadratic response to perturbations and distinguish a necessary pointwise curvature condition from a sufficient functional estimate.
30 min - 07
Jacobi fields and conjugate points
Linearize the stationary equation and use endpoint zeros of its variational equation to detect a loss of second-variation positivity.
32 min - 08
The direct method and weak lower semicontinuity
Prove that a minimizing sequence has an admissible limit whose energy achieves the infimum.
32 min - 09
Relaxation and Young measures
Explain how fine oscillations defeat weak lower semicontinuity and how a relaxed energy records unresolved phase mixtures.
34 min - 10
Variational inequalities and obstacle problems
Derive one-sided optimality for a convex feasible set and solve a one-dimensional contact problem with a verified reaction force.
34 min