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Canonical Variables

Let's write the equation of motion in a generalized form,

$$\ddot{q} \; = \; {1 \over m} \; F$$ (12.7)

where I have used ``q'' as the ``canonical coordinate'' whose second derivative ($\ddot{q}$) 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  $\theta$  [measured in radians], ``m'' is the moment of inertia  I<sub>O</sub>  and ``F'' is the torque  $\Gamma_O$;  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  q_i  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  q_i, we can define the generalized force

$$Q_i = - {\partial V \over \partial q_i}$$ (12.8)

If we then generalize the ``inertial coefficient''  $m \to \mu$,  we can write out   $i^{\rm th}$  equation of motion in the form

$$\ddot{q}_i \; = \; {Q_i \over \mu}$$ (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 . . . . ''