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MathGroup Archive 2002

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5th Degree Polynomials

  • To: mathgroup at smc.vnet.net
  • Subject: [mg37129] 5th Degree Polynomials
  • From: jdhouse4 at mac.com
  • Date: Sat, 12 Oct 2002 05:04:54 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

Hey folks,

I have been working on a problem that seems to not lend itself to a 
solution. The following Mathematica code begins with the expression 
that I am trying to solve. For the curious, it's a degree 2 zonal and 
sectoral harmonics problem where I am trying to calculate and plot the 
geoid of earth as compared to an ellipse to see how well the geoid is 
approximated as an ellipse. In any case, we have the following relation 
ship,

U  =GM/r( 1 - (ae/r)^2 ( J2 (3/2 Sin[t]^2 - 1/2) -  3 Cos[t]^2 (C22 
Cos[2 x] + S22 Sin[2 x]));
Ur =1/2 we^2 (r Cos[t])^2;
W[x_] =U + Ur;

In trying to reorder W to become a function r wrt t, that is r[t_], I 
tried, among others,

Solve[W[t], r]

which returned

({{r ->
         Root[ae^2 GM J2 + 6 ae^2 C22 GM Cos[ [t]] ^2 -
               3 ae^2 GM J2 Sin[ [t]] ^2 + 2 GM #1 ^2 -
               2 W0 #1 ^3 +  we^2 Cos[ [t]] ^2 #1 ^5 &,
           1]}, {r ->
         Root[ae^2 GM J2 + 6 ae^2 C22 GM Cos[ [t]] ^2 -
               3 ae^2 GM J2 Sin[ [t]] ^2 + 2 GM #1 ^2 -
               2 W0 #1 ^3 +  we^2 Cos[ [t]] ^2 #1 ^5 &,
           2]}, {r ->
         Root[ae^2 GM J2 + 6 ae^2 C22 GM Cos[ [t]] ^2 -
               3 ae^2 GM J2 Sin[ [t]] ^2 + 2 GM #1 ^2 -
               2 W0 #1 ^3 +  we^2 Cos[ [t]] ^2 #1 ^5 &,
           3]}, {r ->
         Root[ae^2 GM J2 + 6 ae^2 C22 GM Cos[ [t]] ^2 -
               3 ae^2 GM J2 Sin[ [t]] ^2 + 2 GM #1 ^2 -
               2 W0 #1 ^3 +  we^2 Cos[ [t]] ^2 #1 ^5 &,
           4]}, {r ->
         Root[ae^2 GM J2 + 6 ae^2 C22 GM Cos[ [t]] ^2 -
               3 ae^2 GM J2 Sin[ [t]] ^2 + 2 GM #1 ^2 -
               2 W0 #1 ^3 +  we^2 Cos[ [t]] ^2 #1 ^5 &, 5]}} )

which wasn't too much help, though it is a list of 5 Root functions. 
But in order to plot, I need a function r(t) so I can plot r wrt 
t...right?

ParametricPlot[r[t], {t, 0, Pi}]

So, I guess my questions are as follows:
1. How do I get Solve[ ] to output numbers, as //N and NSolve did 
nothing to Solve[r[t], ...] to get any numbers instead of just r -> 
Root[...]?

2. Is there a way to use ParametricPlot[ W[t], {t, 0.0, Pi}] instead of 
using r[t] and negating the whole issue of solving W[t] for r[t]?

I have read that Solve only works for up to 4th order polynomials. I 
have been unable to find anything that works on my problem, having 
tried SolveAlways[ ] and other, and combination of others.

Any help is welcome. I'll be glad to forward my Notebook if someone 
asks. Thanks ahead of time.

Jim Hillhouse
jdhouse4 at mac.com

Ph.D. Graduate Student
Aerospace Engineering
University of Texas at Austin
jdhouse4 at mail.utexas.edu

512-784-3205



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