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Brachistochrone Equation Problem

  1. Sep 26, 2015 #1
    1. The problem statement, all variables and given/known data
    This is the solution of Brachistochrone .

    upload_2015-9-26_19-47-36.png
    upload_2015-9-26_19-48-17.png

    2. Relevant equations


    3. The attempt at a solution
    I am very confused that how the x in equation(6.24) get its value a(1-cosӨ) ? What is the technique behind this solution of x?
     

    Attached Files:

  2. jcsd
  3. Sep 26, 2015 #2

    mfb

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    Staff: Mentor

    It is just a clever substitution that replaces x. How to find that:
    - in theory: "oh, it is completely obvious that this substitution will lead to a nice solution!"
    - in practice: "let's try several approaches until one of them works"
     
  4. Sep 27, 2015 #3
    Why only a(1-cosӨ ? What is the theory behind this substitution?
     
  5. Sep 27, 2015 #4

    mfb

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    Staff: Mentor

    There is no deeper theory. It is a clever definition of Ө. You can see that it works nicely later.

    You could also choose x=Ө2 + 3 cos(Ө) but that wouldn't help.
     
  6. Sep 27, 2015 #5

    TSny

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    To motivate the substitution (6.24), first make the substitution x = au in (6.23). Then complete the square in the denominator. From your experience with integration, you should then see what might be another good substitution to further simplify the integral.
     
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