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Re: Coordinate Transformations
*To*: mathgroup at smc.vnet.net
*Subject*: [mg19842] Re: Coordinate Transformations
*From*: Jens-Peer Kuska <kuska at informatik.uni-leipzig.de>
*Date*: Sun, 19 Sep 1999 01:20:27 -0400
*Organization*: Universitaet Leipzig
*References*: <7rsitq$3r3@smc.vnet.net>
*Sender*: owner-wri-mathgroup at wolfram.com
Hi Janson,
I assume you have a set of second order differential equations deqn
depending
on the coordinates q[1][t], q[2][t], q[3][t], ...
and wish to transform to the new coordinates Q[1][t], Q[2][t], ...
You can express q[_][t] as functions of Q[_}[t] by the relations
q[k][t]=t[k][Q[1][t],Q[2][t],..] In Mathematica this ist written as
trules={q[1][t]->t[1][Q[1][t],..],..}
since Mathematica does not replace the derivatives you must generate the
rules for the
first and scond derivative by
dtrules=Join @@ ({#,D[#,t],D[#,{t,2}]} & /@ trules)
now you can transform your differential equations with
deqnQ=deqn /. dtrules
for the next coordinate system you can repeat the steps and nest all the
transformations.
Please notice that this not assume that the transformations or the
differential
equations are linear, it works for any set of variable transformations.
For a linear transformation your q=T.Q (q and Q are vectors T is a
matrix)
you get some simple linear combinations.
Hope that
helps
Jens
The future steps depend on your differential equations, because it is
not
Jason Rupert wrote:
>
> Please Reply to: rupertj at email.uah.edu
> Hello All,
> I have crusied the web, read the Mathematica help, and looked at many of the
> posted email questions but none answer my question. Maybe ya'll can help. I
> am working on a project that involves several different coordinate systems.
> These coordinate systems are arbitrarily set-up by me to easyily describe
> that part of the system. I will have to be able to write the position,
> velocity, acceleration, etc. vectors in terms of these several different
> coordinates. All the unit vectors are orthogonal and right-handed. I also
> have all the angles involved in the rotation from one coordinate system to
> another. Is their any way to set-up a rotation matrices so that this can
> easily be done? Does any one have an example that I can follow? Any help you
> can offer will be greatly appreciated. Can you refrence me to any sources to
> examine. Are there any Mathematica user groups where daily postings are
> made?
>
> Thanks again,
> Jason Rupert
> UAH Huntsville, AL
>
> Reply to: rupertj at email.uah.edu
>
> ______________________________________________________
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