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I Numerical Calculus of Variations

  1. Oct 18, 2016 #1
    I attempt to solve the brachistochrone problem numerically. I am using a direct method which considers the curve ##y(x)## as a Lagrange polynomial evaluated at fixed nodes ##x_i##, and the time functional as a multivariate function of the ##y_i##. The classical statement of the problem requires the curve ##y(x)## to have its endpoints on ##(x_0, 0)## and ##(x_1, y_1)## and the initial velocity of the particle to be ##v_0=0##, so that the time functional would take the form

    $$ T[y]=\int_{x_0}^{x_1} \sqrt{\frac{1+\left[\frac{dy}{dx}\right]^2}{2gy}} dx $$.

    The derivative ##\frac{dy}{dx}## is approximated from the Lagrange polynomial and the integral is computed using a quadrature rule, then the functional is minimized using an interior point method. But the problem comes when the denominator of the integrand vanishes, which will always happen at the endpoint ##(x_0, 0)##. I tried to avoid this by setting ##v_0##, but the method fails to converge to a continuous curve for arbitrary small ##v_0##.

    What would be the proper way to handle this situation?
     
  2. jcsd
  3. Oct 20, 2016 #2

    Stephen Tashi

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    Science Advisor

    Intuitively, your formulation makes velocity a function of displacement, so if there is no initial displacement then there "is no reason" the particle should begin moving. You could reformulate the model so velocity depends on acceleration.
     
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