Math 2.0

Calculus of variations and optimal control; optimization Basic notions of the calculus of variations Problem

Isoperimetric problem of the calculus of variations

Builds on Integral functional with a Lagrangian densityAdmissible functionEuler–Lagrange equationLagrange multipliers on Banach spaces

Problem

Let a<ba < b and A,B,ℓ∈ℝA, B, \ell \in \R, and let L,G ⁣:[a,b]×ℝ×ℝ→ℝL, G \colon [a, b] \times \R \times \R \to \R, written L(x,z,p)L(x, z, p) and G(x,z,p)G(x, z, p), be of class C2C^2. Set J(y)=∫abL(x,y,y′) dxJ(y) = \int_a^b L(x, y, y') \, dx and K(y)=∫abG(x,y,y′) dxK(y) = \int_a^b G(x, y, y') \, dx, two integral functionals. The isoperimetric problem asks for a minimiser of JJ among the admissible functions y∈C1([a,b])y \in C^1([a, b]) with y(a)=Ay(a) = A, y(b)=By(b) = B and the integral constraint K(y)=ℓK(y) = \ell.

Remarks

The name comes from the classical problem of the largest area enclosed by a curve of given length, the case where KK is arc length. Dido's problem is a version of it.

The basic necessary condition is Euler's multiplier rule. Let y0∈C2([a,b])y_0 \in C^2([a, b]) be a minimiser, and suppose that y0y_0 does not solve the Euler–Lagrange equation of KK. Then there is λ∈ℝ\lambda \in \R such that y0y_0 solves the Euler–Lagrange equation of L−λGL - \lambda G: ddx(Lp−λGp)=Lz−λGzon [a,b].\frac{d}{dx} \big( L_p - \lambda G_p \big) = L_z - \lambda G_z \quad \text{on } [a, b]. It follows from Lagrange multipliers on Banach spaces with constraint space ℝ\R. If y0y_0 solves the Euler–Lagrange equation of KK, the rule must be replaced by one with multipliers (λ0,λ)≠(0,0)(\lambda_0, \lambda) \ne (0, 0) for λ0L−λG\lambda_0 L - \lambda G, where λ0=0\lambda_0 = 0 is possible.

The catenary problem is an isoperimetric problem: it minimises potential energy at fixed length.

Reading path

To understand this page from the ground up, read these 29 pages in order. The foundations come first.

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Admissible function, concept, no evidence levelAdmissible functionconceptBanach space, concept, no evidence levelBanach spaceconceptComplex numbers, object, no evidence levelComplex numbersobjectComplex plane, concept, no evidence levelComplex planeconceptDual space, concept, no evidence levelDual spaceconceptEuclidean space, object, no evidence levelEuclidean spaceobjectEuler–Lagrange equation, concept, no evidence levelEuler–Lagrange equati…conceptFunction, concept, no evidence levelFunctionconceptIntegral functional with a Lagrangian density, concept, no evidence levelIntegral functional w…conceptKernel of a linear map, concept, no evidence levelKernel of a linear mapconceptReal numbers, object, no evidence levelReal numbersobjectSet bounded above, definition, no evidence levelSet bounded abovedefinitionReal and imaginary parts, definition, no evidence levelReal and imaginary pa…definitionTotal derivative, concept, no evidence levelTotal derivativeconceptFunction of several variables, concept, no evidence levelFunction of several v…conceptSupremum, definition, no evidence levelSupremumdefinitionFréchet derivative, concept, no evidence levelFréchet derivativeconceptLocal maximum and minimum, concept, no evidence levelLocal maximum and min…conceptCompleteness axiom, statement, no evidence levelCompleteness axiomstatementExistence of square roots of nonnegative reals, statement, no evidence levelExistence of square r…statementSquare root, definition, no evidence levelSquare rootdefinitionModulus of a complex number, definition, no evidence levelModulus of a complex …definitionComplex differentiability, definition, no evidence levelComplex differentiabi…definitionDomain in the complex plane, definition, no evidence levelDomain in the complex…definitionHolomorphic function, definition, no evidence levelHolomorphic functiondefinitionOpen mapping theorem, statement, no evidence levelOpen mapping theoremstatementLyusternik's theorem on tangent spaces, statement, no evidence levelLyusternik's theorem …statementLagrange multipliers on Banach spaces, statement, no evidence levelLagrange multipliers …statementIsoperimetric problem of the calculus of variations, problem, no evidence levelIsoperimetric problem…problem
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Admissible function concept
  2. Banach space concept
  3. Complex numbers object
  4. Complex plane concept
  5. Dual space concept
  6. Euclidean space object
  7. Euler–Lagrange equation concept
  8. Function concept
  9. Integral functional with a Lagrangian density concept
  10. Kernel of a linear map concept
  11. Real numbers object
  12. Set bounded above definition
  13. Real and imaginary parts definition
  14. Total derivative concept
  15. Function of several variables concept
  16. Supremum definition
  17. Fréchet derivative concept
  18. Local maximum and minimum concept
  19. Completeness axiom statement
  20. Existence of square roots of nonnegative reals statement
  21. Square root definition
  22. Modulus of a complex number definition
  23. Complex differentiability definition
  24. Domain in the complex plane definition
  25. Holomorphic function definition
  26. Open mapping theorem statement
  27. Lyusternik's theorem on tangent spaces statement
  28. Lagrange multipliers on Banach spaces statement
  29. Isoperimetric problem of the calculus of variations problem

Where this leads

3 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Calculus_of_variations
About this page
Tag6FY0KHMK5A
URIm2://6FY0KHMK5A
Kindproblem
WritingW0 stub
MSC49K05, 49K21
StatusThis copy of the site has no build data. Run m2 status 6FY0KHMK5A.
AboutCalculus of variations
UsesIntegral functional with a Lagrangian density, Admissible function, Euler–Lagrange equation, Lagrange multipliers on Banach spaces

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