Networks in Mechanics

  1. Physics is effective mathematics.
    1. Idea's origin.
          [Gol, p.188, (6-7)] comes from the invariance of the speed of light. However, the definition in [Pet, p.88, l.-13] fails to indicate its relation to the speed of light.
    2. By considering real situations [Gol, p.188, l.-7-p.191, l.5], we can narrow our focus from [Pet, p.88, (2,14)] to [Gol, p.191, (6-15)].
    3. Effective proof (The wave equation is invariant under Lorentz transformations). The proof in [Gol, p.198, l.-10-p.199, l.13] is intuitive. In contrast, Petrovsky's use of reduction to absurdity in [Pet, p.92, l.8-p.94, l.18] shows that he does not understand the theory deeply enough.
    4. In physics, there are more resources (electromagnetic theory [Gol, p.200, l.-3-p.201, l.17]) to establish or check a theory (special relativity). The details of a theory may have outstanding physical meanings [Gol, p.192, l.-12; p.193, l.6].

  2. Going to the basics and the mainstream to increase the opportunities for connection.
        For the definition of canonical transformation, it is more effective to check Fomenko's version [Fom, p.29, Definition 1.8] than Goldstein's version [Gol, p.239, l.7]. Because Fomenko's version is very basic, we may find more of its connections to differential geometry:
    1. The connection between skew-symmetric scalar product and exterior 2-forms [Fom, p.30, l.2-l.14; Arn2, p.221, l.-6-l.-3].
    2. By Stokes' theorem, we may derive Goldstein's version as a theorem [Arn2, p.241, l.25-l.28].

  3. Self-contained books
        A self-contained math book is a book that does not require the reader to refer to other resources. Very often a self-contained book is restricted to one subject and fails to interact with other subjects. This kind of book may give people a wrong impression that a theory can be completed without interacting with other areas. Actually, whether a math book is good or not is determined mainly by how much the theory interacts with other areas. [Col] is a good book because it interacts with physics frequently. In an excellent book, the interaction is very detailed. In other words, even a small step in mathematical reasoning like the chain rule [Rei, p.114, (3.8.8)] has a physical interpretation [Rei, p.113, Fig. 3.8.2]. In contrast, in a poorly written book like [Halm], the entire content is abstract and completely isolated from other areas. Thus, the author of any math book that explains a formal theory but fails to provide interaction with other areas owes his or her readers an apology. It is clear that such an author could not understand how the theory interacts with other areas and left the presentation disconnected.

  4. [Sad, p.305, 8.2.A] establishes [Sad, p.305, (8.2)] as an experimental law, while [Wangs, p.233, 14-5] shows that it is a consequence [Wangs, p.233, (14-30)] of Ampere's law of force between circuits.