Woof Pete, the biggest problem with using words with precise definitions is understanding what they really mean, as seen in that specific context. Maybe we should create a mathematical section on TNS first? As a mere layman I often find the formulas assuming all too much knowledge of what the variables, functions, etc, really mean. They may have been ever so clear to the guy who wrote it, but to us seeing it, a lot of information missing. As for canonical properties this one sums it up quite nicely I think.

"physics can be described as studying how quantities evolve with time, in particular, we are interested in studying those quantities whose measurements can be arrived at by "freezing" time at a particular moment. So the position of the rock that I just threw is one such variable, while its velocity and acceleration are not. Assuming time differentiability, knowledge of the measurements of such quantities is enough: you can differentiate them in time to give the values of other dynamical variables.

To put it more mathematically, physics is the study of a bunch of time dependent functions with are a priori independent from each other.

The goal of physics is then to find rules that associate different such functions: Newton's law of universal gravity assigns a rule of interaction via gravitational attraction between to bodies, and it only suffices to know the positions of those bodies at any given time to find the interaction.

Now, as it turns out, the rules of physics the were formulated based on experimental evidence, in many cases, can be written as differential equations on those measured quantities with second order derivatives in time. Mathematically, this says that physics can be described as the study of a system of time dependent functions whose evolution is governed by a second order ordinary differential equation. (There are, of course, exceptions to this, but in those cases we do not have canonical conjugate variables anyway.)

Now, it is well known that a second order ordinary differential equation can be uniquely solved if we provide, as initial data, the value of the function at time 0, and the value of its first derivative at time 0. This implies the well known Newtonian philosophy that knowing the position and momentum of every particle enables one to solve for their dynamics for all eternity. Now, knowledge of the first derivative is not essential to the knowledge of how the world operates, since with a complete knowledge of the functions for all time will imply knowledge of the first derivatives. Yet, the knowledge of the functions for all time is encoded simply in the knowledge of the function and its first derivative at one particular time and the laws of physics.

Here we have the canonical conjugate variables. We take as half of the variables those time-dependent, a priori mutually independent functions that are sufficient to describe the dynamics. For the other half, we take them to be the time-derivative of those aforementioned functions.

In other words, the canonical conjugate variables are those variables one arrives at from the following procedure in the study of ordinary differential equations:" From

What is a good non-physics definition of canonical conjugate variables? Or "Words used in math and science are often borrowed from the common use. "Given from God" is one of the non-technical meanings of "canonical". Specifically, "appearing in a Biblical canon".

I usually take it to mean "basic" or "simplest", another common (non-technical) meaning of the term. This common usage carries over to meany technical fields. For example, the canonical equation of a circle in Cartesian coordinates is x^2+y^2=r^2 (or maybe even the more basic equation, x^2+y^2=1).

Mathematicians and physicists have yet other meanings: canonical decompositions in math, canonical variables in quantum physics, canonical ensembles statistical physics. Each of these concepts has very precise definitions." By D H..

So, I would suggest using as simple words as possible, if that's impossible you need to rethink the concept and see if you can find another way to describe it. But it also depends on who you want to communicate with. There are some guys here that just luve math

and maybe, a math section would be cool?

But, only if those writing there know their formulas in depth, and actually know how to explain them, equations may look cool but without naming what they describe they're worthless to most of us here. I'm not talking about you here Pete. It's just that I've seen examples of guys going in over their depth at TNS before, bamboozling us with equations not appropriate to what they want to prove/discuss, and not able to explain how they reached their mathematical conclusions either.