Models in Mechanics

  1. Compton's strategy for explaining radiation scattering [Eis, pp.34-40, 2-4].
    1. Simplify the hypothesis.
    2. Use universal principles.
    3. Classify cases in a relevant way.
    4. Explain why classical mechanics fails.

  2. Justification of a model [Eis, p.100, l.2-l.4].

  3. Methods of modeling.
    1. The new model is based on an old model [Cor, p.229, (12-21-(2))]. We modify the old model only where necessary [Cor, p.229, (12-21-(1))].
    2. The infinitesimal đQ is inexact [Rei, p.78, 2.11], but dS=đQ/T [Rei, p.108, (3.6.5)] is exact [Hall, p.409, state variable]. It takes great ingenuity to fit the studied object into the theoretical model of differential forms by converting đQ into dS. This modeling approach not only makes the change of heat more manageable, but also enables us to figure out the true meaning of entropy at last [Rei, p.99, (3.3.12)].

  4. Use the model of sectionally constant potentials to approximate strong nuclear forces [Mer2, p.93, Fig. 6.1].

  5. The assumptions of a macroscopic theory must be justified by the corresponding microscopic theory [Lan7, p.4, l.7-l.17].

  6. The formulas derived from various models must be consistent.
        If ħw is small compared with the band gap for all occupied levels, the formula of AC conductivity for the quantum mechanical model reduces to the formula of AC for the semiclassical model [Ashc, p.253, l.1-l.5].

  7. We adopt the Kronig-Penney model for a 1-dim solid [Eis, p.457, Fig. 13-8] because the model is easier to treat mathematically and it preserves all the important features [Eis, p.457, l.34-l.36] of the real case.

  8. A simpler model easily identifies the direct and essential reason [Rei, p.207, l.20] why the magnetization of paramagnetic material vanishes in zero applied field.
        Note that the model in [Rei, pp.206-208, 6.3, paramagnetism] is simpler than that in [Hoo, 7.2.3, pp.203-205; (7.17)].

  9. Flexibility of a model [Ashc, p.399, l.1-l.6].

  10. The assumptions of a model versus its limits of application [Ashc, p.422, l.1-l.28].
    1. An assumption should not be restrictive [Ashc, p.422, l.17]. That is, it should cover a broad range of cases. If there are exceptions which violate the assumption, we would like to know the main feature of these exceptions.
    2. Sometimes an assumption is not made for general validity, but for analytical necessity. A more complicated case requires another model.

  11. Compromises between the general case and a concrete form.
    1. [Ashc, p.784, footnote 2].

  12. (Point dipole) Idealizing a formula that is valid for large distances so that it becomes valid for all distances.
    1. The real situation: [Wangs, p.114, (8.21)] is only valid for large distances.
    2. Our goal of idealization: [Wangs, p.114, (8.21)] holds everywhere in space.
    3. Justifying the construction of our model: [Wangs, p.119, l.-7-p,120, l.2].

  13. The ideal gas
    1. The thermodynamic model. The model does not have to exactly represent the real case. We idealize the system so that on the one hand it becomes simple and standard, and on the other hand the discrepancy between the ideal case and the real case becomes easily explained (compare [Zem, p. 107, l.-3-l.-2] with [Zem, p.108, (5-6)]).
      The limitations of the model [Zem, p.123, l.10-l.19]
    2. The model in kinetic theory [Zem, 5-10].

  14. How we idealize the real situation and fit it into a model.
    Example. The Maxwell construction [Hua1, p.41, l.-12-p.43, l.11].

  15. Mechanical-electrical analogies [Edw, p.221, Fig. 3.7.3]
        The performance of a mechanical system can be predicted by means of accurate but simple measurements in the corresponding electric model [Edw, p.222, l.2-l.11].