Math 2.0

Calculus of variations and optimal control; optimization Classical problems Problem

Catenary problem

Builds on Isoperimetric problem of the calculus of variationsBeltrami identityHyperbolic cosineArc length

Problem

A uniform, perfectly flexible and inextensible chain of length ℓ\ell hangs under constant gravity from two fixed points (a,A)(a, A) and (b,B)(b, B). Here a<ba < b, and ℓ\ell is larger than the distance between the points. What shape does the chain take?

As a variational problem: among y∈C1([a,b])y \in C^1([a, b]) with y(a)=Ay(a) = A, y(b)=By(b) = B and arc length ∫ab1+y′(x)2 dx=ℓ,\int_a^b \sqrt{1 + y'(x)^2} \, dx = \ell, find one that minimises the potential energy, which is proportional to U(y)=∫aby(x)1+y′(x)2 dx.U(y) = \int_a^b y(x) \sqrt{1 + y'(x)^2} \, dx.

Remarks

This is an isoperimetric problem. With a multiplier λ\lambda, the Lagrangian is (y−λ)1+y′2(y - \lambda) \sqrt{1 + y'^2}, which does not depend on xx. The Beltrami identity gives (y−λ)/1+y′2=c(y - \lambda) / \sqrt{1 + y'^2} = c, and the solutions with c>0c > 0 are the catenaries y(x)=λ+ccosh⁡(x−dc),y(x) = \lambda + c \cosh\Big(\frac{x - d}{c}\Big), where the constants cc, dd, λ\lambda are fixed by the two end conditions and the length. Here cosh⁡\cosh is the hyperbolic cosine. Jacob Bernoulli posed the problem in Acta Eruditorum, and solutions by Leibniz, Huygens and Johann Bernoulli were published in 1691.

Reading path

To understand this page from the ground up, read these 34 pages in order. The foundations come first.

Show the path as a graph
Admissible function, concept, no evidence levelAdmissible functionconceptArc length, concept, no evidence levelArc lengthconceptBanach 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 levelFunctionconceptHyperbolic cosine, concept, no evidence levelHyperbolic cosineconceptIntegral 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…conceptPartial derivative, concept, no evidence levelPartial derivativeconceptCompleteness axiom, statement, no evidence levelCompleteness axiomstatementExistence of square roots of nonnegative reals, statement, no evidence levelExistence of square r…statementBeltrami identity, statement, no evidence levelBeltrami identitystatementSquare 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…problemCatenary problem, problem, no evidence levelCatenary 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. Arc length concept
  3. Banach space concept
  4. Complex numbers object
  5. Complex plane concept
  6. Dual space concept
  7. Euclidean space object
  8. Euler–Lagrange equation concept
  9. Function concept
  10. Hyperbolic cosine concept
  11. Integral functional with a Lagrangian density concept
  12. Kernel of a linear map concept
  13. Real numbers object
  14. Set bounded above definition
  15. Real and imaginary parts definition
  16. Total derivative concept
  17. Function of several variables concept
  18. Supremum definition
  19. Fréchet derivative concept
  20. Local maximum and minimum concept
  21. Partial derivative concept
  22. Completeness axiom statement
  23. Existence of square roots of nonnegative reals statement
  24. Beltrami identity statement
  25. Square root definition
  26. Modulus of a complex number definition
  27. Complex differentiability definition
  28. Domain in the complex plane definition
  29. Holomorphic function definition
  30. Open mapping theorem statement
  31. Lyusternik's theorem on tangent spaces statement
  32. Lagrange multipliers on Banach spaces statement
  33. Isoperimetric problem of the calculus of variations problem
  34. Catenary problem problem

Further reading

  1. Web https://en.wikipedia.org/wiki/Catenary
About this page
Tag8REBFQD6E0
URIm2://8REBFQD6E0
Kindproblem
WritingW0 stub
MSC49K05, 49K21, 49-03
StatusThis copy of the site has no build data. Run m2 status 8REBFQD6E0.
AboutCalculus of variations
UsesIsoperimetric problem of the calculus of variations, Beltrami identity, Hyperbolic cosine, Arc length

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