Re: Instantaneous axis of rotation?
- From: Henning Makholm <henning@xxxxxxxxxxx>
- Date: Thu, 04 May 2006 22:05:13 +0200
Scripsit "Edward Green" <spamspamspam3@xxxxxxxxxxx>
Well, this definition of w seems reasonable, does it not. One problem
-- the reference ray was arbitrary! We have described the intantaneous
rate of rotation about an arbitrary body fixed axis, not "the"
instantaneous axis of rotation, should such a thing exist. What
singles out a particular axis when apparently any one will do?
The orthodox mathematical construction goes something like:
Imagine a coordinate system that is fixed in space and one rigidly
attached to the tumbling body. For each point in time, write down
the transformation that goes from the fixed coordinates to tumbling
ones. It will have the general form
X |-> T_t + M_t X
where T_t is a translative term (which we will ignore) and M_t is
a matrix in SO(3).
Let t=0 be the time at which we want to find the instantaneous
rotation axis (and angular velocity). We can have chosen to attach the
tumbling coordinate system in such a way that at t=0 it happens to be
oriented just as the fixed one is, i.e. M_0 = I.
Since everything happens smoothly M_t will have a time derivative
N = dM/dt such that for an infinitesimal dt,
M_dt = I + dtN_0.
It we calculate what it means for I+dtN_0 to be orthogonal (setting
dt²=0) ge get that N_0 must be an antisymmetric 3×3 matrix. That
means that it has just three independent components, and if we work
through all the algebra of imagining a different orientation for the
_fixed_ coordinate system, we find that those three components of N_0
transforms under such rotations as a pseudovector. That this works is
an accident of 3D geometry; it does not work out that way in other
dimensions.
We now see that we can choose to turn the lab coordinates such that
only one of the three components of N_0 is nonzero, e.g.
( 0 -a 0 ) ( 1 -adt 0 )
N_0 = ( a 0 0 ) M_dt = ( adt 1 0 )
( 0 0 0 ) ( 0 0 1 )
and we then find that the points fixed by M_dt are exactly the ones
on the z-axis. (There is nothing magical about this, because our last
rotation of the lab frame was determined by the movement). Therfore
the direction the z frame ended up in is the instantaneous axis of
rotation of the body at t=0.
In fact, using the matrix exponential, t -> exp(tN_0) defines a
constant rotation about this axis which at t=0 matches the movement of
the original body to the first degree in t. We see that a is the
instantaneous *angular velocity* of the rotation.
Hmm... body fixed axis... body fixed axis... maybe there's my problem.
We don't want to describe the rotation about an arbitrary axis fixed in
the body, we want to identify an axis _not_ fixed to the body, in
general wandering around the body, which is the best of all possible
axes which the body can be said to be spinning around for that moment.
But how? And what is the mathematical expression of "best" here?
The fundamental property is that the rotation axis consist of all
those points that are momentarily at rest relative to the center of
the body. The mathematical development above consitutes a proof sketch
that those points always form a well-defined geometrical line through
the center - except in the degenerate case that N_0=0, in which case
the body is momentarily non-rotating.
--
Henning Makholm Blessed are those with no
shoes, for the earth shall kiss them.
.
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