Math 2.0

Calculus of variations and optimal control; optimization Classical problems Problem

Euler's elastica

Builds on Curvature of a curvePlane curveIsoperimetric problem of the calculus of variationsPendulum equationElliptic function

Problem

Let ℓ>0\ell > 0. Consider plane curves γ ⁣:[0,ℓ]→ℝ2\gamma \colon [0, \ell] \to \R^2 of class C2C^2, parametrised by arc length. Prescribe the end points γ(0)\gamma(0), γ(ℓ)\gamma(\ell) and the unit tangents γ′(0)\gamma'(0), γ′(ℓ)\gamma'(\ell). Among such curves, find those that make the bending energy E(γ)=12∫0ℓκ(s)2 dsE(\gamma) = \frac{1}{2} \int_0^\ell \kappa(s)^2 \, ds stationary, where κ\kappa is the signed curvature of γ\gamma. The solutions are the elasticae, the shapes of a thin elastic rod held at its ends.

Remarks

Write γ′(s)=(cos⁡θ(s),sin⁡θ(s))\gamma'(s) = (\cos \theta(s), \sin \theta(s)), so that κ=θ′\kappa = \theta'. Let (X,Y)=γ(ℓ)−γ(0)(X, Y) = \gamma(\ell) - \gamma(0). The end points give the constraints ∫0ℓcos⁡θ ds=X\int_0^\ell \cos \theta \, ds = X and ∫0ℓsin⁡θ ds=Y\int_0^\ell \sin \theta \, ds = Y. So this is an isoperimetric problem with two constraints. With multipliers, the Euler–Lagrange equation becomes θ′′+csin⁡(θ−θ0)=0\theta'' + c \sin(\theta - \theta_0) = 0 for constants cc and θ0\theta_0, the equation of a pendulum. So the solutions are expressed through elliptic integrals and elliptic functions. Euler classified the possible shapes of the elasticae in 1744.

Reading path

To understand this page from the ground up, read these 69 pages in order. The foundations come first.

  1. Admissible function concept
  2. Banach space concept
  3. Complex numbers object
  4. Complex plane concept
  5. Curvature of a curve concept
  6. Differential equation concept
  7. Dual space concept
  8. Elliptic function concept
  9. Euclidean space object
  10. Euler–Lagrange equation concept
  11. Function concept
  12. Integral functional with a Lagrangian density concept
  13. Interval concept
  14. Kernel of a linear map concept
  15. Plane curve concept
  16. Real numbers object
  17. Set bounded above definition
  18. Euler's formula statement
  19. Real and imaginary parts definition
  20. Ordinary differential equation definition
  21. Total derivative concept
  22. Function of several variables concept
  23. Lipschitz condition definition
  24. Grönwall's inequality statement
  25. Supremum definition
  26. Order of a differential equation definition
  27. Fréchet derivative concept
  28. Local maximum and minimum concept
  29. Solution of an ordinary differential equation definition
  30. Completeness axiom statement
  31. Autonomous system definition
  32. Initial value problem definition
  33. Existence of square roots of nonnegative reals statement
  34. Linear differential equation of order n definition
  35. Equilibrium point definition
  36. Square root definition
  37. System of linear differential equations definition
  38. Homogeneous linear equation definition
  39. Linear equation with constant coefficients definition
  40. Uniqueness of solutions for Lipschitz vector fields statement
  41. Modulus of a complex number definition
  42. Homogeneous linear system with constant coefficients definition
  43. Maximal solution definition
  44. Superposition principle statement
  45. Characteristic equation definition
  46. Complex differentiability definition
  47. Matrix exponential as the solution of a linear system statement
  48. Domain in the complex plane definition
  49. Flow of a vector field definition
  50. Solution space of a homogeneous linear equation has dimension n statement
  51. Holomorphic function definition
  52. Lyapunov stability definition
  53. Trajectory definition
  54. Fundamental system of solutions definition
  55. Open mapping theorem statement
  56. Asymptotic stability definition
  57. First integral definition
  58. Solutions for distinct real roots statement
  59. Stability of linear systems via eigenvalues statement
  60. Solutions for repeated roots statement
  61. Lyusternik's theorem on tangent spaces statement
  62. Linearisation at an equilibrium concept
  63. Lagrange multipliers on Banach spaces statement
  64. Solutions for complex conjugate roots statement
  65. Hyperbolic equilibrium point definition
  66. Harmonic oscillator example
  67. Isoperimetric problem of the calculus of variations problem
  68. Pendulum equation example
  69. Euler's elastica problem

Further reading

  1. Web https://en.wikipedia.org/wiki/Elastica_theory
About this page
TagCBT8F8S8JT
URIm2://CBT8F8S8JT
Kindproblem
WritingW0 stub
MSC49K05, 74K10, 53A04
StatusThis copy of the site has no build data. Run m2 status CBT8F8S8JT.
AboutCurvature of a curve
UsesCurvature of a curve, Plane curve, Isoperimetric problem of the calculus of variations, Pendulum equation, Elliptic function

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