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Effectiveness in Mechanics

  1. Proof of [H, L]=0.
    Ineffective method: [Hec2, p.31, (79)].
    Effective method: [Coh, p.779, l.3-l.9].


  2. An advanced textbook should discuss points in greater depth. In [Jack, pp.95-96, 3.1], we assume the potential has the form of [Jack, p.95, (3.2)]. It seems that the discussion of the solutions which do not have the form  is left out. However, [Lev2, p.124, l.9] shows that in most cases of physical interest this product form includes all the solutions [Lev2, pp.124-125, (6.12)-(6.16)].

  3. Using symmetry allows us to reduce mechanical calculations [Sad, p.104, l.3-l.5; p.126, l.7-l.12].
    1. The circular symmetry in [Sad, p.114, Fig. 4.7; p.270, Fig. 7.8(a)] allows us to avoid evaluating the integral of the first term in [Sad, p.115, (4.24); p.270, l.10] (see [Sad, p.115, l.6-l.10; p.270, l.13]).
    2. Choosing a cylindrical surface symmetric with respect to both z-axis and xy-plane makes the calculations in [Sad, p.127, 4.6.B] more effective than those in [Sad, pp.112-114, 4.3.A].
    3. Using Gauss's (Ampere's) law to find the electric (magnetic) field.
          Given symmetric charge (current) distributions, it is possible to find an integral surface (path) over which the electric (magnetic) field is constant in magnitude.
      Examples. [Sad, pp.126-132, 4.6; pp.274-281, 7.4].


  4. Suppose we begin with an effective algorithm [Hei, p.25, (5.5); p.36, l.9-p.37, l.-20; p.38, l.1-l.21]. Then we give some definitions along the way so that we have some terminology to refer back to as we proceed. Without the effective algorithm, the introduced definitions would be meaningless and the theory would be baseless.
        Heine introduces the concepts "group representation [Hei, p.26, l.13-16], a basis [Hei, p.28, l.-14] and reducibility [Hei, p.33, (5.16)]" using the same example given in [Hei, p.25, l.-11; p.26, l.-16] as a guide. Thus, Heine's introduction is natural and effective. In contrast, Tinkham introduces the concepts "group representation [Tin, p.18, l.-5] and reducibility [Tin, p.20, l.4]" in abstract settings. The method that Tinkham uses to derive the irreducible representations from his example [Tin, p.19, l.9-l.21] fails to use basic functions and thus is not effective as Heinr's. Without an effective algorithm as the foundation, his introduction is artificial and leaves unsolved the problem of how to find effective methods to check the definitions  using a basis.