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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_O  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 . . . . "