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How to Estimate with a Computer program a 3rd order diff eq

  1. Apr 13, 2009 #1
    How to Estimate with a Computer program a 3rd order differential equation with 4 variables and 4 equations? The equations are derived from the geodesics equation and the Schwarzchild metric. The variables are r, theta, phi and t. Any hints would be appreciated.
     
  2. jcsd
  3. Apr 14, 2009 #2
    Probably if you can write down your equations it will be much clearer what your problem actually is.
     
  4. Apr 14, 2009 #3
    Equation 1:
    [tex]\frac{d}{d\tau}(\frac{2r}{r-2m}(\frac{dr}{d\tau})) + \frac{2m}{(r-2m)^{2}}(\frac{dr}{d\tau})^{2} - r(\frac{d\theta}{d\tau})^{2} - rsin^{2}\theta (\frac{d\phi}{d\tau})^{2} + \frac{mc^{2}}{r^{2}}(\frac{dt}{d\tau})^{2} = 0[/tex]

    Equation 2:
    [tex]\frac{d}{d\tau}(r^{2}\frac{d\theta}{d\tau}) - r^{2}sin\theta cos\theta (\frac{d\phi}{d\tau}) = 0[/tex]

    Equation 3:
    [tex]\frac{d}{d\tau}(r^{2}sin^{2}\theta (\frac{d\phi}{d\tau})) = 0[/tex]

    Equation 4:
    [tex]\frac{d}{d\tau}(\frac{r-2m}{r}(\frac{dt}{d\tau})) = 0[/tex]

    where m and c are constants
     
    Last edited: Apr 15, 2009
  5. Apr 14, 2009 #4
    It look like your equations have analytical solutions (probably i can be wrong). Start solving equation 4 first, then equation 3.
    If I'm not mistaken, for the Schwarzchild metric, there is analytic solution.

    But if you still insist on numerical solution you can try Rungge-Kutta fourth order. One of the easiest to program. But you need to rewrite your equations as system of first order DE of the form.

    [tex]X'(\tau) = A(\tau) X(\tau)[/tex]

    where X is a column vector and A is a matrix.


    By the way why do you say your equation is 3rd order?
     
  6. Apr 14, 2009 #5
    Yes the equation is 2nd order not 3rd.
     
    Last edited: Apr 15, 2009
  7. Apr 15, 2009 #6
    Just checking my math, in the first part of Equation 2:

    [tex]\frac{d}{d\tau}(r^{2}\frac{d\theta}{d\tau}) = 2r\frac{dr}{d\tau}\frac{d\theta}{d\tau} + r^{2}(\frac{d\theta}{d\tau})^{2}[/tex]

    Is this correct?
     
  8. Apr 15, 2009 #7
    I think from equation 2 and 3 you can obtained much simplier equation, I think.


    [tex]\frac{d}{d\tau}(r^{2}\frac{d\theta}{d\tau}) = A \cot (\theta) [/tex]

    where A is the arbitrary constant from eq. 2 :biggrin:
     
  9. Apr 16, 2009 #8
    Any suggestions on how to solve for r and [tex]\theta[/tex] from the first two equations if the solution from equation 3 is :

    [tex]\frac{d\phi}{d\tau} = \frac{constant3}{r^{2}sin^{2}\theta}[/tex]

    and equation 4 is:

    [tex]\frac{dt}{d\tau} = \frac{(constant4)r}{r-2m}[/tex]
     
    Last edited: Apr 16, 2009
  10. Apr 16, 2009 #9
    I was reading this post and find it an intresting questing although general relativity is not my field :-) Finding an analytic solution seems to be a challenge and that's what I would like to try. Now it seems that there are 5 variables involved, [itex]\tau[/itex], [itex]r[/itex], [itex]\theta[/itex], [itex]\phi[/itex] and [itex]t[/itex]. Can someone tell me which variable is depending on what?
     
  11. Apr 19, 2009 #10
    Look like tau is the independent variable to me. But what confuse me is that both the letters tau and r look similar in latex.
     
  12. Apr 19, 2009 #11
    tau takes the place of s in geodesic equations and is called proper time in a "local free-fall frame replacing s by:

    [tex]ds^{2} = -c^{2}d\tau^{2}[/tex]
     
    Last edited: Apr 19, 2009
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