Math 2.0

Calculus of variations and optimal control; optimization Classical problems Problem

Dido's problem

Builds on Isoperimetric problem of the calculus of variationsIsoperimetric inequalityPlane curve

Problem

Let ℓ>0\ell > 0. Consider plane curves of length ℓ\ell that start and end on a fixed straight line and otherwise lie on one side of it. Which of them encloses, together with the line, the largest area?

In coordinates, the line is the xx-axis and the curve is a piecewise C1C^1 map t↦(x(t),y(t))t \mapsto (x(t), y(t)), 0≤t≤10 \le t \le 1, with y(0)=y(1)=0y(0) = y(1) = 0 and y≥0y \ge 0. When the curve has no self-crossings, the enclosed area is ∣∫01y(t) x′(t) dt∣\abs{\int_0^1 y(t) \, x'(t) \, dt}.

Remarks

The answer is a semicircle of length ℓ\ell, with its diameter on the line; it encloses the area ℓ2/(2π)\ell^2 / (2\pi). Reflecting a curve in the line turns the problem into the classical isoperimetric problem for closed curves of length 2ℓ2\ell. So the answer follows from the isoperimetric inequality in the plane. Written with such a parametrised curve, it is an isoperimetric problem of the calculus of variations. Its Euler–Lagrange equation with a multiplier says that the curve has constant curvature, so it is an arc of a circle. The name comes from the legend of the founding of Carthage. Dido was granted as much land as an ox hide could enclose; she cut the hide into a thin strip and laid it out against the coast.

Reading path

To understand this page from the ground up, read these 39 pages in order. The foundations come first.

Show the path as a graph
Admissible function, concept, no evidence levelAdmissible functionconceptBanach space, concept, no evidence levelBanach spaceconceptBorel sigma-algebra, concept, no evidence levelBorel sigma-algebraconceptCompact set, concept, no evidence levelCompact setconceptComplex 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 mapconceptOuter measure, concept, no evidence levelOuter measureconceptParametrised surface, concept, no evidence levelParametrised surfaceconceptPlane curve, concept, no evidence levelPlane curveconceptReal numbers, object, no evidence levelReal numbersobjectSet bounded above, definition, no evidence levelSet bounded abovedefinitionReal and imaginary parts, definition, no evidence levelReal and imaginary pa…definitionLebesgue measure, object, no evidence levelLebesgue measureobjectTotal derivative, concept, no evidence levelTotal derivativeconceptFunction of several variables, concept, no evidence levelFunction of several v…conceptSurface area, concept, no evidence levelSurface areaconceptSupremum, definition, no evidence levelSupremumdefinitionBrunn–Minkowski theorem, statement, no evidence levelBrunn–Minkowski theor…statementFréchet derivative, concept, no evidence levelFréchet derivativeconceptLocal maximum and minimum, concept, no evidence levelLocal maximum and min…conceptCompleteness axiom, statement, no evidence levelCompleteness axiomstatementIsoperimetric inequality, statement, no evidence levelIsoperimetric inequal…statementExistence 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…problemDido's problem, problem, no evidence levelDido's problemproblem
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. Borel sigma-algebra concept
  4. Compact set concept
  5. Complex numbers object
  6. Complex plane concept
  7. Dual space concept
  8. Euclidean space object
  9. Euler–Lagrange equation concept
  10. Function concept
  11. Integral functional with a Lagrangian density concept
  12. Kernel of a linear map concept
  13. Outer measure concept
  14. Parametrised surface concept
  15. Plane curve concept
  16. Real numbers object
  17. Set bounded above definition
  18. Real and imaginary parts definition
  19. Lebesgue measure object
  20. Total derivative concept
  21. Function of several variables concept
  22. Surface area concept
  23. Supremum definition
  24. Brunn–Minkowski theorem statement
  25. Fréchet derivative concept
  26. Local maximum and minimum concept
  27. Completeness axiom statement
  28. Isoperimetric inequality statement
  29. Existence of square roots of nonnegative reals statement
  30. Square root definition
  31. Modulus of a complex number definition
  32. Complex differentiability definition
  33. Domain in the complex plane definition
  34. Holomorphic function definition
  35. Open mapping theorem statement
  36. Lyusternik's theorem on tangent spaces statement
  37. Lagrange multipliers on Banach spaces statement
  38. Isoperimetric problem of the calculus of variations problem
  39. Dido's problem problem

Further reading

  1. Web https://en.wikipedia.org/wiki/Isoperimetric_inequality
About this page
TagFABJSZ9D3S
URIm2://FABJSZ9D3S
Kindproblem
WritingW0 stub
MSC49K05, 52A40, 49-03
StatusThis copy of the site has no build data. Run m2 status FABJSZ9D3S.
AboutIsoperimetric inequality
UsesIsoperimetric problem of the calculus of variations, Isoperimetric inequality, Plane curve

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