Re: Re: Eulerian angles
- To: mathgroup at smc.vnet.net
- Subject: [mg42701] Re: [mg42682] Re: [mg42668] Eulerian angles
- From: "Marko Vojinovic" <vojinovi at panet.co.yu>
- Date: Tue, 22 Jul 2003 04:40:38 -0400 (EDT)
- References: <200307190720.DAA17034@smc.vnet.net> <200307201021.GAA29283@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
I don't know how to create a graphic using Mathematica (maybe somebody else
on the list can help you with that), but I can give you these instructions
on how to draw them on paper:
1) Draw a right-handed orthogonal coordinate system Oxyz, and another one
(also orthogonal right-handed) Ox'y'z' which is rotated in an arbitrary way
with respect to the first. Point O is the common origin.
2) Draw the line which is the intersection of the planes Oxy and Ox'y'. That
line is sometimes called "nodus" or "nodal line", denoted n.
3) The angle xOn, between the x-axis and the nodal line, is the first Euler
angle. It is usually called "precession", and denoted \psi (in LaTeX
notation...).
4) The angle x'On, between the x'-axis and the nodal line, is the second
Euler angle. It is called "rotation", and denoted \phi (of course, one may
use any other notation as well).
5) The angle zOz', between the z-axis and the z'-axis, is the third Euler
angle. It is called "nutation", and denoted \theta.
That's the simplest way to visualize these angles (that I know of). If you
wish, I can send you formulae which connect coordinate basis vectors e_x,
e_y, e_z to the basis e_x', e_y', e_z' in terms of Euler angles.
There is also a simple procedure how to rotate one coordinate frame into
another by rotating around respective axes of these three angles (though it
is very simlpe --- you can even guess it once you draw the picture).
Physicists are familiar with this stuff for a number of reasons. Euler
angles are useful in (particularly simple way of) describing the motion of a
rigid body in Newtonian mechanics, are used as parameters when dealing with
SU(2) and
SO(3) symmetries (and their Lie algebra) in paticle physics, relativity
theory and elsewere where something is "rotationally invariant" (like the
physics of Hydrogen atom and so on), and generally some PDEs can be
simplified and solved in terms of them.
Best regards,
Marko
================================================================
Murphys computer law: 34. There's never time to do it right, but always time
to do it over.
================================================================
> On Sat, 19 Jul 2003 03:20:01 -0400 (EDT), Selwyn Hollis
> <selwynh at earthlink.net> wrote:
>
> > Some 5 or 6 years ago, I asked a question in MathGroup about the "Euler
> > angles" that are used by RotateShape. Apparently physicists know all
> > about this stuff, but I still have almost no feeling for what these
> > angles are about. So I thought I'd issue this challenge:
> >
> > Create *the* graphic illustrating the Euler angles that ought to be in
> > the Mathematica Book --- hopefully understandable by a hack
mathematician
> > and his calculus students.
> >
> > The winner will receive glowing praise and thanks in a soon-to-be
> > published book.
> >
> > -----
> > Selwyn Hollis
> > http://www.math.armstrong.edu/faculty/hollis
> >
> >
- References:
- Eulerian angles
- From: Selwyn Hollis <selwynh@earthlink.net>
- Re: Eulerian angles
- From: Dr Bob <drbob@bigfoot.com>
- Eulerian angles