[yor_on]

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. 

[Bill S (OP)]

There are some guys here that just luve math and maybe, a math section would be cool?

A math section would be cool; but the coolest type would be one that made the maths accessible to "hitch-hikers" like me.

[JP]

Back to the original topic, I have something that might help (or confuse) you, Bill. 

I don't want to post another site's image here, so you can click on this link to see what I'm going to describe:  http://jeb.biologists.org/content/214/22/3829/F2.large.jpg  What you want to focus on are the top row of plots and the bottom row (we'll get to the middle later).  The top row shows a sound wave vs. time, which you're probably familiar with.  The height of the wave indicates the amplitude (energy) and the horizontal axis indicates time, so you have a plot of the energy of the sound wave vs. time.  The bottom row is what's called a power spectrum. 

It turns out you can generate the top plots by adding up a bunch of plots of sine waves, where a different amount of each sine wave is added.  Since we know each sine wave has a specific frequency, we can make a plot of how much of each frequency sine wave needs to be added to get the top plots.  This is what's being shown on the bottom plot.  The height of the plot indicates the amount (amplitude) of each sine wave, and the horizontal axis indicates the frequency of the sine wave.  By doing this transformation, we've gone from position space (top plots) to frequency space (bottom plots), where frequency space basically tells us the amount of each sine wave needed to construct the original waveform.

The same thing holds for position/momentum space.  The math gets worse, since position is usually 3 dimensional, and so momentum is as well.  Now instead of just frequency, the sine waves have a frequency (1 dimension) and direction (2 dimensions).  But you can do the same trick and figure out how much of each sine wave you add to get the original waveform.  If you plot this, it will be a plot of the amount (amplitude) of each sine wave vs. frequency and direction.  For a sine wave, the frequency is the magnitude of momentum and the direction is the direction of momentum.  So momentum space is just a way of saying which sine waves make up the original wave. 

-------------------------------

I'm not sure if that's clear or not, but its a hand-wavy way of getting at the mathematics relating time to frequency and position to momentum.

The middle plots in the above link are another way of writing a function in terms of position and momentum (actually time and frequency in that plot).  The horizontal axis is time, the vertical axis is frequency, and the color indicates "how much energy" there is at that time for that frequency.  These kinds of plots are called phase space plots, and they're extremely useful.  If you're interested, I can go on at length about their properties, since these are one of my research interests.  But this is a long post so I'll leave you with this: I'm sure you're familiar with one kind of phase space plot of frequency (vertical axis) vs. time (horizontal), which is a way of transcribing a sound wave into something more usable by musicians:



[Source of the last image: http://en.wikipedia.org/wiki/Sheet_music]

[imatfaal]

Back to the original topic, I have something that might help (or confuse) you, Bill. 
/snipped

Helped me!  Nice post JP !

[Pmb]

"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.

Can I send you an article by Karl Popper?

[yor_on]

Sure

[Bill S (OP)]

Back to the original topic, I have something that might help (or confuse) you, Bill.

I'm doing my best to make it help, but I'm easy to confuse.

[JP]

Back to the original topic, I have something that might help (or confuse) you, Bill.

I'm doing my best to make it help, but I'm easy to confuse.

I always find a glass of nice single malt helps with that. 

[Geezer]

I actually understood this bit! (The words too.)

[Source of the last image: http://en.wikipedia.org/wiki/Sheet_music]

[imatfaal]

Back to the original topic, I have something that might help (or confuse) you, Bill.

I'm doing my best to make it help, but I'm easy to confuse.

I always find a glass of nice single malt helps with that. 

I find that a nice glass of single malt helps with practically anything

[Bill S (OP)]

On the subject of single malt, I was introduced to a new (to me) one at Christmas.  I suppose I shouldn’t mention the name, but, as usual the description is interesting.

Of the nose, it says:  “Smells delightfully of fruitcake, banana nut bread and sweet malt in the first aromatic go-rounds; seven minutes of further exposure to the air brings out devilishly toasty/roasted aromas of crispy pork rind, sweet oak, vanilla, red grapes and blackberry jam”.

And the taste?  “Entry is vividly sweet and grainy, yet nimble in its textural weight and acidic agility; gains momentum by mid-palate as the taste profile suddenly bursts with red fruit presence (red currants, mostly) and oaky vanilla.  Finishes clean as a whistle, chewy and concentrated.  Tight, composed, no body fat”.

I guess I’m a philistine, but I couldn’t find the blackberry jam.

[yor_on]

There should be tar and some seagulls lonely calls in it too.

That the best single malt to me
And no hangovers..

[Pmb]

Sure

Hi yor_on,

I just realized what a difficult task that would be to scan in all those pages. Perhaps I'll do it someday. In the mean time, what I wanted to convey is located here http://plato.stanford.edu/entries/popper/#SciKnoHisPre

Don't get the idea that I understand all that. I've just started to read Popper's work. That started a while back when I bought The Logic of Scientific Discovery by Karl Popper. I also have a philosophy of science book that I'm now reading. This stuff is hard reading my friend.

[yor_on]

Yeah, I've looked at Popper before and he seems like his works demands a quite extensive reading. I remember his thoughts as having a lot to do with QM, but it is some time ago. Thanks for the link

[yor_on]

This

Popper's experiment.