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How difficult is it to solve this elliptic ODE.

  1. Nov 29, 2013 #1
    Hello, I am working on a research problem and I am not sure whether or not I will be able to figure this out in a suitable amount of time. I have never solved a single elliptic integral and they do seem non-trivial to gain an understanding of (most of the books I've glanced at assume a very high level knowledge of differential equations). I do not need an exact solution to this integral(I am in the regime where 1/r^3 <<1), but I am just curious how difficult it would be to obtain an approximate solution to r(λ) for

    [itex] (\frac{dr}{dλ})^2 = E^2 - \frac{L^2}{2r^2} + \frac{GML^2}{r^3} [/itex]

    This is an important result for physics, namely it is a radial null geodesic in the Schwarzschild metric. Usually substitutions are made to bring it to a differential equation of dr/d(phi), but I want to know the solution in terms of the affine parameter λ.

    I tried separating, solving the radial coordinate and then back-solving for λ, but this was very far from being useful. Mathematica is not much help in solving this ode sadly, but I do recognize that it should have a solution in terms of elliptic functions. I even tried a u=1/r substitution and did not gain any ground that way. Any thoughts or tips on good resources and how difficult this should be to solve are welcomed.
     
  2. jcsd
  3. Nov 29, 2013 #2

    ShayanJ

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    Gold Member

  4. Nov 30, 2013 #3
    Hi !
    it's a separable ODE. So the function λ(r) can be expressed on integral form (attachment)
    Then, an approximate is obtainded on the form of a limited series expansion. For less deviation, expand it with more terms.
    If you need an approximate of the function r(λ), you have to inverse the series.
     

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