BELIEVE ME NOT! -
- A SKEPTICs GUIDE
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Let's write the equation of motion in a generalized form,
![\begin{displaymath}\ddot{q} \; = \; {1 \over m} \; F
\end{displaymath}](img20.gif) |
(12.7) |
where I have used ``q'' as the ``canonical coordinate''
whose second derivative (
)
is the ``canonical acceleration.''
Normally q will be the spatial position x
[measured in units of length like metres or feet],
but you have already seen one case
(rotational kinematics) in which ``q''
is the angle
[measured in radians],
``m'' is the moment of inertia IO
and ``F'' is the torque
;
then a completely analogous set of equations pertains.
This turns out to be a quite common situation.
Can we describe simply how to go about formulating
the equations of motion for ``systems'' that might even be
completely different from the standard objects of
Classical Mechanics?
In general there can be any number of canonical coordinates qi
in a given ``system'' whose behaviour we want to describe.
As long as we have an explicit formula for the
potential energy V in terms of one or more qi,
we can define the generalized force
![\begin{displaymath}Q_i = - {\partial V \over \partial q_i}
\end{displaymath}](img24.gif) |
(12.8) |
If we then generalize the ``inertial coefficient''
,
we can write out
equation of motion in the form
![\begin{displaymath}\ddot{q}_i \; = \; {Q_i \over \mu}
\end{displaymath}](img27.gif) |
(12.9) |
which in most cases will produce a valid and workable solution.
There is an even more general and elegant formulation
of the canonical equations of motion which we will discuss
toward the end of this chapter.
I am not really sure how the term canonical
came to be fashionable for referring to this abstraction/generalization,
but Physicists are all so fond of it by now that
you are apt to hear them using it in all their conversations
to mean something like archetypal:
``It was the canonical Government coverup . . . '' or
``This is a canonical cocktail party conversation . . . . ''
Next: Differential Equations
Up: ``Solving'' the Motion
Previous: Timing is Everything!
Jess H. Brewer
1998-10-08