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Extremizing functionals (Calculus of variations)

  1. Feb 22, 2009 #1
    1. The problem statement, all variables and given/known data

    Find the curve y(x) that extremizes the functional J[y]= int({1-y'^2}/y,x=a..b) if the end points lie on two non-intersecting circles in the upper half-plane.

    2. Relevant equations

    Euler's equation: if F=F(x,y,y') then Euler's equation extremization is found from solving the ODE: F_y - d/dx(F_y') = 0 (where F_y denotes the derivative of F wrt y).

    Since F is independant from x the problem reduces to solving y'*F_y' -F= c

    3. The attempt at a solution

    I used the Euler's equation to solve the given functional.

    At some point I get: y'^2 = c*y -1
    so if i want to solve for y' I will have to choose wether to give it + or - sign (first confusion).

    Suppose the first problem is trivial. I got the extremum to be a parabola with 2 constants.


    Now my main problem is to use the boundary conditions to determine the curve y(x).
    So y(a) will be varying on the first circle and y(b) on the second.
    I tried to read in my text (Gelfand and Fomin) but I couldn't just solve it.

    Any help :)?

    Thanks,
    Joe
     
  2. jcsd
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