Math 2.0

Calculus of variations and optimal control; optimization Classical problems Problem

Minimal surface of revolution

Builds on Integral functional with a Lagrangian densityBeltrami identitySurface areaSurface of revolutionCatenoid

Problem

Let a<ba < b and A,B>0A, B > 0. Among curves y∈C1([a,b])y \in C^1([a, b]) with y>0y > 0, y(a)=Ay(a) = A and y(b)=By(b) = B, find one whose surface of revolution about the xx-axis has the least surface area S(y)=2π∫aby(x)1+y′(x)2 dx.S(y) = 2\pi \int_a^b y(x) \sqrt{1 + y'(x)^2} \, dx.

Remarks

The Lagrangian y1+y′2y\sqrt{1 + y'^2} does not depend on xx, so the Beltrami identity gives y/1+y′2=cy / \sqrt{1 + y'^2} = c for a constant c>0c > 0. The solutions are the catenaries y(x)=ccosh⁡((x−d)/c)y(x) = c \cosh\big((x - d)/c\big), and their surfaces of revolution are catenoids. Depending on the positions of the two end points, there are two, one or no catenaries through them. When the end circles are far apart compared with their radii, no catenary joins the points and there is no minimiser among such smooth curves. Then the infimum of SS is π(A2+B2)\pi(A^2 + B^2), the area of the two end discs, approached by curves that run close to the axis; this limit is the Goldschmidt solution. For some positions where catenaries do exist, the two discs still have smaller total area than every catenoid through the points, and again no smooth curve attains the infimum.

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Catenoid, example, no evidence levelCatenoidexampleEuclidean 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…conceptParametrised surface, concept, no evidence levelParametrised surfaceconceptSurface of revolution, example, no evidence levelSurface of revolutionexampleFunction of several variables, concept, no evidence levelFunction of several v…conceptSurface area, concept, no evidence levelSurface areaconceptPartial derivative, concept, no evidence levelPartial derivativeconceptBeltrami identity, statement, no evidence levelBeltrami identitystatementMinimal surface of revolution, problem, no evidence levelMinimal surface of re…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.
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  1. Catenoid example
  2. Euclidean space object
  3. Euler–Lagrange equation concept
  4. Function concept
  5. Integral functional with a Lagrangian density concept
  6. Parametrised surface concept
  7. Surface of revolution example
  8. Function of several variables concept
  9. Surface area concept
  10. Partial derivative concept
  11. Beltrami identity statement
  12. Minimal surface of revolution problem

Further reading

  1. Web https://en.wikipedia.org/wiki/Minimal_surface_of_revolution
About this page
TagAEG6QWEFRG
URIm2://AEG6QWEFRG
Kindproblem
WritingW0 stub
MSC49Q05, 49K05, 53A10
StatusThis copy of the site has no build data. Run m2 status AEG6QWEFRG.
AboutMinimal surface
UsesIntegral functional with a Lagrangian density, Beltrami identity, Surface area, Surface of revolution, Catenoid
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