- #1
The-herod
- 24
- 0
First of all, I'm sorry about the last topic, accidentally I switched between the previous message and this one... Sorry about the troubles. I think it's the right forum (after reading a bit), sorry if I'm wrong...
My high school graduation project is about the perturbation theory, it's application in free fall and a comparison between the efficiency of several methods. The first method I'm learning is Poincare's method, and there are a few points I don't understand. I'll be much thankful if you could help me with it, or reference me to some textbook, article or book that explains it nice and clear (I couldn't find one...).
So we start with
[tex]{\frac{\mathrm{d}\mathbf{x}}{\mathrm{d}t} = \mathbf{F}(\mathbf{x},t,\epsilon)[/tex]
We say that
[tex]\mathbf{x}(t,\epsilon)=\sum_{k=0}^\infty \mathbf{x}_{(k)}(t)\cdot\epsilon^k[/tex]
We substitute the last equation in the first one.
I know it's a lot of question, so as I said in the beginning, I'll much appreciate any reference to any good resource.
Thanks a lot! (and sorry for the long message)
My high school graduation project is about the perturbation theory, it's application in free fall and a comparison between the efficiency of several methods. The first method I'm learning is Poincare's method, and there are a few points I don't understand. I'll be much thankful if you could help me with it, or reference me to some textbook, article or book that explains it nice and clear (I couldn't find one...).
So we start with
[tex]{\frac{\mathrm{d}\mathbf{x}}{\mathrm{d}t} = \mathbf{F}(\mathbf{x},t,\epsilon)[/tex]
We say that
[tex]\mathbf{x}(t,\epsilon)=\sum_{k=0}^\infty \mathbf{x}_{(k)}(t)\cdot\epsilon^k[/tex]
We substitute the last equation in the first one.
- Now comes the confusing part. Why do we compare between the coefficient of the same power of epsilon on both sides? It's just one solution of the equation, isn't it? I mean, theoretically, there are also other solutions for this equation, and we do it just for the comfort, to get a simple ODE of one variable for each x(k), am I wrong?
- The second question is - what is, exactly, epsilon? I know it's a small parameter, and that the accuracy of the solution after we find the nth first x(k) is [tex]O(\epsilon^n)[/tex], so the smaller epsilon is, the greater the accuracy of the solution is. So why not just choose epsilon=0 (it gives us the equation we had in the beginning)?
Which leads me to the last and most basic question... - Why do we do these all from the beginning? We only get multiple ODEs instead of one, so how does it help us? If we know how to solve the equation for x(k), we'll probable know how to solve also the first equation, for x, don't we?
Am I missing anything...? (I guess I am...)
I know it's a lot of question, so as I said in the beginning, I'll much appreciate any reference to any good resource.
Thanks a lot! (and sorry for the long message)