I'd like some suggestions for improving this article, and would also like to get some help with expanding the references section! HappyCamper 22:21, 10 Apr 2005 (UTC)
First list of suggestions
Disclaimer: I am not a mathematician. Nevertheless, I have read the article with pleasure. It is an article on a subject that is of general use in mathematics and physics. Some points to consider:
- The section that introduces substitutions uses χ (chi) as a substition for x. Later on, q is silently used. This may be confusing readers.
- Unfortunately, wikipedia does not seem to provide good definitions of a force function, transient and steady state solutions to link to
- I would present the example of nondimensionalization of a first order differential equation even earlier in the article, where the 5 steps are explained. I suggest to move the more general case of linear differential equations to after the discussion of the second order equations.
- In this more general case of linear differential equations, I did not understand why this system behaves as a mixture of first and second order systems. Following some links did not help: the page on characteristic polynomial talks about matrices and the contents at superposition were also not very helpful.
- The section on the universal second order linear oscillator presents the transient solution and derives the steady-state solutions. This does not seem very relevant to the topic of nondimensionalization. Instead, it seems more relevant to the harmonic oscillator article. Should it be moved there?
- The discussion on RC and RL circuits would be clearer if it concentrated on one instead of two circuits.
- All examples in the text deal with differential equations and the introduction suggest that this is the main application. Are there other applications?
Jan van Male 21:04, 12 Apr 2005 (UTC)
First list of suggestions feedback
- Thanks for your feedback - I really appreciate it! Here are my responses...
- When I wrote the article, I couldn't decide whether I should use q consistently or not. I realize that the switchover from using the Greek χ to q might be confusing. However, I wanted to emphasize that it doesn't really matter what letter is picked for the substitution. I'll add something to the article that recommends, for example, something that might sound like this: "To nondimensionalize a variable, usually a convenient symbol is used to represent the new dimensionless quantity. For example, if the original problem involves length represented with the letter x, it would be judicious to represent the new nondimensionalized variable with l (for length), or χ (reminds us that it is related to "x"), or some other appropriate place holder. The choice should be made so that it is easy to keep track of the variables, although it is largely a personal choice of the person performing the nondimensionalization." --> I'll try to keep this comment short though, as I feel that this is something that applies to all mathematical problems in general - Choose variable names that make intuitive sense to make thinking through the problem easier!
- Actually, come to think of it, forcing function, transient state and steady state really don't need to be mentioned in nondimensionalization at all. They are only there because of the particular examples I picked to show how nondimensionalization can be applied for problems involving those things. I might try to phase these terms out somehow. What do you think?
- I did not put an example of nondimensionalization involving the first order system near the numbered steps because nondimensionalization applies to other things as well, in addition to differential equations. However, differential equations are probably the most illustrative example of applying the technique, so maybe it would be best to change this section as you suggest. It would be nice to keep the continuity of the 5 points though, as its intention was to give someone reading this article an idea of the thought process that is involved. As for the more general case, I think it would be best if I relegated this to another article. Jump to last point too...
- I'm very glad you pointed this out...this is in fact a critical error I overlooked! I was implicitly assuming certain things which I didn't mention, in particular, that all the coefficients are assumed to be real. This section will definitely need a rewrite. I might decide to relegate this section to the characteristic polynomial page instead. I only mentioned this in the article to demonstrate the importance of understanding nondimensionalization of first and second order systems. Understanding the behaviour of these these two is sufficient to help understand all linear DEs with constant coefficients because of the superposition principle.
- Hmm...I wasn't aware of the harmonic oscillator article. I'm glad you pointed this out! I'll move these derivations over to that page - I feel that they're more appropriate there too. What's really important is just to keep the nondimensionalized second order differential equations here, and also the "characteristic units" derived based on the differential equation. They show how the resonant frequency and linewidth come about naturally in the equation.
- I might even move these over to an appropriate circuits page, as what I really wanted to show was that nondimensionalization can recover the time constant of these circuits, and that the time constant does not come out of thin air as an aribtrary definition.
- Nondimensionalization is not restricted to differential equations, but they are probably the most illustrative of the examples. This is because many physics problems (especially those involving movement) can be formulated in terms of differential equations. Maybe I should mention this at the beginning of the article? What do you think? The other cases where nondimensionalization could be used is probably too trivial to illustrate here. The complexities introduced by nondimensionalization for these cases would be hardly necessary. I guess it's safe to say right at the top of the article that this technique is very well suited for simplifying differential equations. I think this a fair statement given the material that is covered in the article.
- So, in summary, here are the changes that I think would make the article better:
- Mention that the choice of variable name substitution is arbitrary, but should be chosen so they make intuitive sense in the context of the problem being solved;
- Expand on related topics on other pages so that the article does not lose its continuity when a user clicks on a wikified link;
- Perhaps phase out terminology which does not directly apply to nondimensionalization;
- Rewrite or move the section about how the general linear DE can be understood by a superposition of smaller systems. The information presented there is not completely correct and needs to be fixed;
- Move derivations of the nondimensionalized transient and steady state to harmonic oscillator;
- Illustrate one circuit example at a time instead of all at once;
- Put the first order example right by the 5 points so it becomes more illustrative.
These were great recommendations. Thanks again for going through the article. I'm glad you enjoyed it! As for the others reading through this, please add your comments to this page, and feel free to help improve the article! HappyCamper 01:16, 13 Apr 2005 (UTC)
Consistent use of variables is good
- Thank you for your kind words. I agree with all of your responses, except for one: I think it is a bad idea to use different symbols for the same concept in a single article. Yes, the choice is arbitrary and mentioning this won't hurt at all. However, once the choice is made, stick to it within a single article. My guess is that people who do not realize that the actual choice of variable names is arbitrary, will probably not understand much of the article anyway. I think that this class of readers will not be enlightened by a change in variable names in this article. In contrast, there is the possibility of a reader being temporarely confused because of different symbols and I think we should try to avoid that. Jan van Male 01:07, 14 Apr 2005 (UTC)
- What if we replaced all the variables not representing time (like position, charge, voltage, and magnetic flux) with χ when nondimensionalized, but also mention right at the substitution what χ is meant to represent? That way, the notation is consistent, and also the meaning would be clear in the context where it's being used. Would this be a good choice for clarity? Well, I'll give the change a try and see how it turns out... :) HappyCamper 02:59, 14 Apr 2005 (UTC)
Update on changes
More of the changes have been put in place. I added a note about mechanical, electrical, torsional (twisting systems), caloric (heat systems) and fluidic systems in the article somewhere. Is it necessary to show that all characteristic units of these systems can be derived from nondimensionalization? Maybe it would be nice to make a table that shows the equivalence between these sytems, but I think this is work for another article.
I changed the electrical oscillations section. It seems to be a bit dominant in the article, and perhaps introducing a "biased point of view" to the article :) --> What to do about this? Or is it okay?
Also, if someone could find a nice nonlinear differential equation to nondimensionalize, that would be a great addition to this article! HappyCamper 00:57, 15 Apr 2005 (UTC)
- I think it would be a mistake to show all characteristic units of the various systems you mention. These characteristics belong in the articles that deal with these systems in more detail. I'd say that these articles can simply list the result of nondimensionalization and link to the nondimensionalization page for an explanation. I agree that showing the equivalence between different systems belongs to a different article. I'd say this can easily become a huge list: the list of phenomena that can be described with first order DEs alone is huge or would there be some way to make this a finite list?
- Regarding the electrical oscillations, I think that the variants of these systems belong to the corresponding parent articles. I'd say that the nondimensionalization article is best served with a few (say two or three) examples that are explained in detail. Ideally, these examples are as different as possible in mathematical complexity and physical meaning but I realize it may be hard to come up with suitable examples. Jan van Male 13:26, 15 Apr 2005 (UTC)
- Well, I started another article called system equivalence. If you're up for it, maybe we can work on it together...But for now, it'll be nice to get this article into a state where the content is pretty stable. For linear DEs, only 2 examples are sufficient to illustrate all the possibilities -> one 1st order, and 2nd order. I'd like to keep the 2nd order mechanical one, since that's one of the few systems I can think of where we could actually have an animated picture for in the article (I put up a request for a mass-spring-damper system somewhere). I'll move some of the electrical examples to another article. HappyCamper 16:13, 15 Apr 2005 (UTC)