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Piece wise smooth function

  1. Jan 28, 2016 #1
    • Member warned about posting without the homework template
    Hello
    Here is my question
    Capture.110PNG.PNG
    So I solved Euler DE and find
    gif.gif
    and when we apply the boundary condition we obtain y=0 . My teacher said that we should write it as two different function as
    matrix%7D%20x%20%26%200%3Cx%3CH%5C%5C%20-x+1%20%26%20H%3Cx%3C1%20%5Cend%7Bmatrix%7D%5Cright.gif
    where H is (1/2).He solved this equation with this way
    gif.gif

    So Here are my questions:
    a) why don't we accept the y=0 as our desire function?
    b)how can obtain (a) constant in my equation with Euler approach?because in multidisciplinary function we can't find a ?
     
  2. jcsd
  3. Jan 28, 2016 #2

    Ray Vickson

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    Science Advisor
    Homework Helper

    What is a "multidisciplinary function"? I have never heard that term.

    Anyway, is the subject matter one of Calculus of Variations, for ##J = \int_0^1 (1-y'^2)^2 dx##? If so, ##F_{y'} = -4 y'(1-y'^2) = -4 y' + 4 y'^3##, so the Euler equation gives ##F_{y} = 0 = (d/dx) F_{y'}##, or ##(3 y'^2- 1) y'' = 0##, so either ##y'' = 0## or ##y' = \pm 1/ \sqrt{3}##.

    On the other hand, we can change variables to ##y'= z##, to get the problem
    [tex] \min \:K(z) = \int_0^1 (1-z^2)^2 \, dx [/tex]
    with no specified boundary conditions on ##z(0), z(1)##. Without using the Euler-Lagrange equation at all we see that we can make ##K(z) = 0## by taking ##z(t)^2 = 1## for all ##t##, and that gives ##x'(t) = \pm 1##, as your instructor said. Certainly that is an optimal solution, since ##K(z) \geq 0## for all PWS functions ##z(t)##, and ##K(z) = 0## is attained by the solution where ##z(t)^2 = 1 \; \forall \, t##.

    The interesting thing about this problem is that the so-called necessary conditions of Euler and Lagrange give the wrong solution. In other words, the Euler-Lagrange equations fail! I suspect that the reason lies in some violation of the hypotheses that underlie the Euler-Lagrange derivation (and which are normally not stated or are ignored when solving problems).
     
    Last edited: Jan 28, 2016
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