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

[Philosophaie]

Main Question or Discussion Point

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.

 

[matematikawan]

Probably if you can write down your equations it will be much clearer what your problem actually is.

 

[Philosophaie]

Equation 1:
$$\frac{d}{d\tau}(\frac{2r}{r-2m}(\frac{dr}{d\tau})) + \frac{2m}{(r-2m)^{2}}(\frac{dr}{d\tau})^{2} - r(\frac{d\theta}{d\tau})^{2} - rsin^{2}\theta (\frac{d\phi}{d\tau})^{2} + \frac{mc^{2}}{r^{2}}(\frac{dt}{d\tau})^{2} = 0$$

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

Equation 3:
$$\frac{d}{d\tau}(r^{2}sin^{2}\theta (\frac{d\phi}{d\tau})) = 0$$

Equation 4:
$$\frac{d}{d\tau}(\frac{r-2m}{r}(\frac{dt}{d\tau})) = 0$$

where m and c are constants

 

[matematikawan]

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.

$$X'(\tau) = A(\tau) X(\tau)$$

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


By the way why do you say your equation is 3rd order?

 

[Philosophaie]

Yes the equation is 2nd order not 3rd.

 

[Philosophaie]

Just checking my math, in the first part of Equation 2:

$$\frac{d}{d\tau}(r^{2}\frac{d\theta}{d\tau}) = 2r\frac{dr}{d\tau}\frac{d\theta}{d\tau} + r^{2}(\frac{d\theta}{d\tau})^{2}$$

Is this correct?

 

[matematikawan]

I think from equation 2 and 3 you can obtained much simplier equation, I think.


$$\frac{d}{d\tau}(r^{2}\frac{d\theta}{d\tau}) = A \cot (\theta)$$

where A is the arbitrary constant from eq. 2

 

[Philosophaie]

Any suggestions on how to solve for r and $$\theta$$ from the first two equations if the solution from equation 3 is :

$$\frac{d\phi}{d\tau} = \frac{constant3}{r^{2}sin^{2}\theta}$$

and equation 4 is:

$$\frac{dt}{d\tau} = \frac{(constant4)r}{r-2m}$$

 

[coomast]

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, $\tau$, $r$, $\theta$, $\phi$ and $t$. Can someone tell me which variable is depending on what?

 

[matematikawan]

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.

 

[Philosophaie]

tau takes the place of s in geodesic equations and is called proper time in a "local free-fall frame replacing s by:

$$ds^{2} = -c^{2}d\tau^{2}$$