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Michael Schwartz
12-13-2000, 03:09 AM
I cannot resist adding my thoughts (perhaps there
should be an entire listserv dedicated to inertial
forces).

My thoughts are not unique, but rather adapted
(stolen) from the brilliant work

"The Variational Principles of Mechanics" by
Cornelius Lanczos (Dover Inc., New York).

I would strongly urge anyone involved in mechanics
to read as much of this as possible. It is fairly
dense material but gives amazing insight. In Ch.
IV, Lanczos deals exclusively with d'Alembert's
Principle. When I first saw this, I was
confused. I had always thought (been taught) that
this "principle" was merely algebraic shuffling.
However, it is noted by Lanczos that "it is
exactly this apparent triviality which makes
d'Alembert's principle such an ingenious invention
and at the same time so open to distortion and
misunderstanding.".

This current "debate" is a great example of how
prophetic Lanczos was!!!

The most important point is that by transforming a
non-equilibrium problem to an equilibrium problem
by the addition of the inertial forces, the
Principal of Virtual Work now holds (the virtual
work of all forces vanishes). This then means
that the PVW and all of it's consequuences can be
directly applied to a "dynamic" problem.

One of these consequences, related to gait
analysis, is to allow the use of "kinematical
variables", that is, velocities that are not the
derivatives of actual position coordinates. The
example cited by Lanczos is the spin of a top
about its axis of symmetry, however one could
extend this to think of the angular velocities of
Euler angles.

Finally, d'Alembert's principle makes clear the
exact origin and nature of the different apparent
forces arising from moving reference frames
whether in translation, rotation or change in rate
or direction of rotation (mass * omega_dot X R),
which Lanczos dubs the "Euler force".

Well that's my $0.02 as we say on this side of the
pond. Thanks to all contributors for the
stimulating discussion.

--
Michael Schwartz, Ph.D.

Director of Bioengineering Research
Gillette Children's Specialty Healthcare

Assistant Professor
Orthopaedic Surgery, Biomedical Engineering
University of Minnesota

Phone:(651)229-3929 Fax:(651)229-3867

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