[Newbie:] How to discard higher orders of variables?

On Wed, Jul 11, 2007 at 09:46:12PM +0200, Schirmacher, Rolf wrote:
> Hello Daniel,

Hello Rolf, 

Thanks for the more detailed description of your problem. I am
actually quite interested in this type of problem (I have recently
been doing some fluid mechanics problems involving similar algebraic
simplification issues), but I am not very familiar with thin shell
equations specifically, nor do I have your references available.

It is extremely helpful to use maxima for reproducing other's
algebra. or even for inventing your own so I welcome you to the group!

> So, what would the "real world" look like? I have attached a wxMaxima input
> file as a starting point. Everything looks quite simple so far. The next
> step would be to loosen the n:0 restriction and generate cut-on frequency
> formulas for arbitrary n. These formulas are given in the literature, but at
> the moment, reproducing them includes some manual substitutions for me. You
> might try, it is not that bad...
> Even more interesting would be a general solution of the det=0 equation
> (might be bound to n=0 for the moment) for kappa, i.e. the dispersion
> equation. All you currently find in the literature (as far as I know) are
> plots and analytical approximations for small Omega (limited to something
> like Omega < 0.05 for n = 1). But be carefull, the terms of the complete
> solution will fill quite some screens on your computer... So, simplifying /
> approximating the det before solving for kappa should help ...


> So, I hope, this is not too boaring to you and perhaps you might have some
> helpful hints?

I've taken a look at your file, and given that I have only a very
limited idea of what you're trying to do, I have some observations as
follows:


kappa can be solved directly by maxima using solve(det,kappa) but it
produces, as you say, many many terms in some of the solutions. The
use of "ratsimp(det,kappa)" shows that the equation is an 8th order
polynomial, so it should have 8 solutions. I'm impressed that maxima
can find them all, given the general nontractability of 5th and higher
order polynomials. However, perhaps some of these expansive solutions
are uninteresting?  As I say, I am not familiar with the specifics of
the theory. Two of the solutions for kappa are quite simple, but I
don't know how physically relevant they are.

If I assume that kappa, Omega, beta are all smallish, and that
therefore products of high powers of these are relatively unimportant,
then I can simplify quite a bit...

ratsimp(det);
ratsubst(0,Omega^3,%);
ratsubst(0,beta^3,%);
ratsubst(0,beta^2*kappa^6,%);
solve(%,kappa);

gives some almost tractable solutions... 

kappa = 0 or +- a largish expression...

if I assume in this expression that beta^2*Omega^2 = 0 I get:

                                                2       2
(%o109) [kappa = - Omega sqrt(- ((4 nu - 6) beta  + 2 nu  + nu - 3)
                2        3       2                 2     3     2
/((nu - 1) Omega  + (3 nu  - 3 nu  - 4 nu + 4) beta  + nu  - nu  - nu + 1)),
                                     2       2
kappa = Omega sqrt(- ((4 nu - 6) beta  + 2 nu  + nu - 3)
                2        3       2                 2     3     2
/((nu - 1) Omega  + (3 nu  - 3 nu  - 4 nu + 4) beta  + nu  - nu  - nu + 1)),
kappa = 0]

Good luck though, those were a lot of assumptions... Let us know how
it goes.



-- 
Daniel Lakeland
dlakelan at street-artists.org
http://www.street-artists.org/~dlakelan