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  • More on inertia?

    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

    Phone651)229-3929 Fax651)229-3867

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