Systematization Schemes for Mechanics and Concept Maps

Ajit R. Jadhav's picture

1. Introductory

Recently, there has been some active discussion on topics like:
-- Open-source textbooks
-- Comparing lecture notes
-- Unification of mechanics
-- Wikipedia and Citizendium

In general, the discussion is on what experts could do to make the learning of mechanics easier, better, more cost-effective, more widely available, etc.

Let me add my two cents.

First of all, I think that books are here to stay. The available books may not fulfill all the expectations of all the people. Yet, the books are written by many of the same people who might participate in projects such as those mentioned above. In writing book, each author can bring his own individual flavor to the teaching of the same material. Further, authors can and do vary in terms of selection of topics, depth, rigor, etc. This is valuable, it is how it should be. Afterall, there are so many differences in lecture notes too--whether put out for free on the Internet or not.

So, I would say that any idea to replace books would not be very successful.

What, however, can possibly be done by the community as a whole is to provide the materials/resources/means to let the reader integrate his understanding of mechanics.

Broadly speaking, this requires systematization of material more than writing of new material.

The systematization can be done along three different lines: (i) by systematically pointing out the differences in knowledge arising in different contexts, (ii) by systematizing precisely how special knowledge is relvant in multi-disciplinary problems, and (iii) by systematically arranging material as per mathematical commonality.

2. Contextual Differences

The meaning of a concept is determined by its context. This is especially true of the predominantly mathematical concepts such as those used in an applicable science like mechanics.

By pointing out how the same topic can be treated differently in different contexts, both enrichment and long-range integration of
knowledge become facilitated.

As of today, this is seldom, if at all, done. The only scheme ever put forth is that of multi-level simulation or scale-dependent organization of materials science. But the characteristic size of phenomenon is only one way to organize material. Consider further examples.

For instance, consider what the topic: "gyroscope" would mean in different contexts of knowledge:

-- philosophy of science: e.g., whether gyroscopic action lends credence to Mach's theory or not, to theologic assertions or not, etc.;
-- physics: see the treatment in, e.g., Goldstein's book on Classical Mechanics;
-- mathematics: say, as a special application of some abstract theory such as that involving coupling or linear systems;
-- engineering mechanics: see any introductory text-book, e.g. Shames or Beer & Johnston;
-- aerospace engineering: say, concerning the details of design, response-time, stability, reliability etc..

Similarly, consider how engineering mechanics people now routinely use the ideas that were first formulated, brought forth, or solved by quantum physicists: matrix formulations of linear systems, eigen-value problem, variational calculus, perturbation theory, etc.

If you browse through a mathematical physics text and an engineering text on any of these topics, not just the notation and the level of rigor but also the difference of conceptual emphasis becomes inescapable.

Similarly, the topics that have recently moved from mathematical physics to engineering would be: chaos i.e. dynamic instability, nonlinear differential equations, difference equations, transform methods, qualitative studies of differential equations, catastrophe theory, etc.

Thus, the conclusion is that the same topic can be seen from many different perspectives, some more fundamental and long-range, others more pertaining to practical objectives in the here and now.

Now, if, for each important topic or principle of mechanics, if there existed a Web-based resource to systematize and readily provide different hyper-links to the contextual differences in a systematic way, it would be great--it would not only help enrich knowledge but also provide a helpful guide-map.

3. Multi-disciplinary Problems

Another, complementary, way to systematize the knowledge resource would be to take a whole series of some concrete applications and point out precisely how the different topics of mechanics come have a relavance in each such a case.

Just as an example, consider the jet-engine turbine blade. One can think of providing links to the existing or new material written in the fields of: (i) solid mechanics (design, models for creep and fracture, etc.) (ii) materials science (structure-property-performance-design relations) (iii) thermal sciences (e.g. transients in heat) (iv) fluid mechanics angle (e.g. boundary layers, momentum transfer, CFD, etc.) I have left out many other disciplines not directly related to mechanics, e.g. NDT, corrosion engineering, etc.

This could be another way to organize knowledge. It would be particularly useful to let industry people see for themselves how even basic research in mechanics has important benefits.

4. Mathematical Commonality

Traditionally, this has often been the favorite method of organizing knowledge in mechanics and physics. For example, one can take the Poisson equation and show how it comes up in electrostatics, soap bubbles, heat transfer, and so on and so forth.

However, as the above two approaches indicate, this is not the only method to systematize knowledge.

5.  Concept Maps

Here is a description of concept maps:

The idea of concept maps is not, really speaking, a different approach to systematize knowledge. Concept maps only provide a means to abstractly depict the result of a systematization effort. The systematization itself would have been accomplished using approaches like those mentioned above.

The main function of concept maps is to make it easy to realize the inter-relations among many different topics and thereby facilitate conceptual integration. This is their fundamental use.

Concept maps would also facilitate communications among different experts when they work together on a common forum (e.g. iMechanica).

Finally, I would like to mention that the experience of mathematicians in organizing or systematizing their field could come in handy. For more information of this example, see:

(Some of iMechanica members possibly are closely associated with AMS too!)

However, please note, due to the abstract nature of mathematical sciences, the first approach discussed above, namely, the differences arising due to context, is not a major issue in mathematics, and the second issue, namely multi-disciplinary problems is by definition absent. Therefore, one could, perhaps, at least entertain the possibility of drawing the entire concept map of mathematics on a 2D paper. However, that kind of simplicity is just not possible for a field like mechanics. For example, consider the material selection diagrams given in Ashby's well-known papers and books, starting (I guess) from his formulation of the deformation-mechanism maps. Ashby's diagrams are nothing but a particular kind of concept maps. They are concept maps as arranged in a quantitatively specified 2D space, whereby each point in the space has a special significance and the significance is limited to its value measured along the two axes. This is not necessarily the case for concept maps in general. Note the many different depictions possible in just one area of mechanics, and you would appreciate the multi-level complexity of the mechanics-related topics.

6. To Conclude

Overall, I believe, the abovementioned systematization would automatically achieve the goal of unifying mechanics. It would also help in bringing experts of widely different pursuations together. It can be an effort in public-domain, and surely would help future students.