Unification in Mechanics

  1. A single piece.
    1. [Gol, p.203, l.-14-l.-11].

  2. How to unify two concepts: Locating soft spots (like g(u) in [Rob, p.7, l.11]) for possible development.
        Both the Galilean transformation and the Lorentz transformation come from the same theory [Rob, p.6, l.9-p.8, l.8]. Only their additional assumptions are different: The former assumes that the speed of light is infinite [Rob, p.8, l.9], while the latter assumes that the speed of light is a constant<+¥ [Rob, p.8, l.19].

  3. The same idea with different interpretations. [Gol, p.257, l.4] relates the Poission bracket of functions to the Lie bracket of vector fields [Spi, vol.1, p.212, l.-3].

  4. The same phenomenon viewed from different reference frames. Example. [Cor, p.293, l.-2-p.294, l.2].

  5. Invariance under different reference frames. Example. Maxwell's equations: The first pair [Cor, p.314, l.-16] & the second pair [Cor, p.314, l.-14].

  6. Unification [Go2, p.554, l.9-l.-15] often requires that we adopt a revolutionary viewpoint to smooth the transition. Examples: [Sym, p.313, l.-14-l.-12] & [Go2, p.553, l.-12-l.-10].

  7. To unify Halmilton's principle, Fermat's principle and Maupertuis' principle of least action requires broad knowledge [Born, p.719, Appendix I].
    1. These principles are theorems [Born, p.129, §3.3.2] rather than axioms.
    2. [Born, p.127, (1)] is the optical version of [Born, p.736, (85)].
      Fermat's principle is the optical version of the principle of least action [Born, p.742, l.5].
    3. Optics is a more refined subject than mechanics in the following sense:
      1. The methods in optics are more effective than those in mechanics. For example, the concept of the Fresnel zones provides an effective algebraic method to calculate the electric field [Hec, p.488]. However, in the general electromagnetic theory we have to use the analytic method of solving Maxwell's equations.
      2. For any formulation in mechanics there corresponds a significant equivalent in optics [Born, p.734, l.8]. However, for a certain formulation in optics, the corresponding equivalent has no significant meaning in practice [Born, p.739, l.5]. However, in the case of diffraction, the concept of diffraction in optics does lead to the discovery of electron diffraction [Born, p.744,l.-2].
    4. Correctness and clarity are key for impressing a true student of physics. If a reader has to spend a lot of time clarifying the confusion and correcting the mistakes of a book, then any honor that the book's author received will not help increase the value of his work.
          [Born, p.722, l.20-l.24] should be corrected as follows:
      Here by "transversality" we mean that the direction of the normal field (U, V, W) of the surface S(x,y,z)=S1 coincides with the direction (dx, dy, dz) of the extremal of the field.

  8. Only through comparing various forms of a cross section may we understand the essence of the concept. That is, only after shedding the nonessential parts may the key point reveal itself.
        There are four forms of a cross section:
    1. (A fixed solid angle; one scatterer) A beam of identical particles passes through a central-field [Coh, p.906, (A-3)].
    2. (A fixed angle of deflection; one scatterer) A beam of identical particles pass through a central-field [Lan1, p.49, (18.15)].
    3. (A fixed angle of deflection; many scatterers) A beam of charged particles is shot through a thin foil [Sym, p.138, (3.273)].
    4. The collision between two beams [Lan2, p.34, l.-7].
            Although the various forms make one dizzy, the essential message is the same. (A) and (B) are the same. (B) is a special case of (C). (D) can be considered the total cross section of (C).

  9. (Variation of the constants). [Col, p.15, l.10]
        [Eis, p.N-3, (N-20) & (N-21)], we may write the solution of [Eis, o.N-3,(N-19)] as [Eis, p.N-3, (N-22)]. By [Eis, p.I-2, (I-8) & (I-9)], we may write the solution of [Eis, p.I-2, (I-7)] as [Eis, p.I-3, (I-11)]. The idea behind the above method is similar to variation of the constants [Col, p.15, l.3].
        Using the method of variation of the constants, we obtain the solution directly by integration. Using [Eis, p.I-3, (I-11)], we reduce [Eis, p.I-2, (I-7)] to [Eis, p.I-3, (I-12)], which is solvable by means of the power series technique. Using [Eis, p.N-3, (N-22)], we reduce [Eis, p.N-2, (I-19)] to [Eis, p.N-3, (I-23)], which is solvable by means of the power series technique.
    Remark. The physical considerations [Eis, p.I-2, (I-8), | u| ® ¥ ; p.N-3, l.-15, r ® ¥ ] help us reduce the differential equations to the desired “homogeneous” form.

  10. It seems that [Rei, §6.2] & [Rei, §6.4] deal with different problems. It turns out that we can use the system-reservoir approach [Rei, p.212, l.1] to solve both problems.
        It seems that [Rei, §6.4] & [Rei, §6.10] study the same problem and derive the same canonical distribution [Rei, p.212, (6.4.2) & p.231, (6.10.13)] but via different approaches. [Rei, §6.4] separates the ensemble into two parts (a system and a reservoir) and then maximizes the entropy. In contrast, [Rei, §6.10] considers the ensemble as a whole and then maximizes the number of the possible configurations [Rei, p.231, l.10]. However, it turns out that the two approaches are equivalent [Rei, p.231, (6.10.15)].

  11. [Rei, §1.5], [Rei, §1.11], [Rei, §3.3], [Rei, §3.7], [Rei, §6.2], [Rei, §6.7], the two methods in [Rei, §6.8] and [Rei, p.347, l.8-l.12] all illustrate the following point: If N is large, the function in discussion attains its maximum at a specific point with a narrow dispersion. This fact has the following consequences:
    1. Only a few summands contribute appreciably to the sum [Rei, p. 17, Fig.1.4.1; p.111, (3.7.17); p.222, l.14; p.347, (9.6.4)].
    2. Only a small region of integration contributes appreciably to the integral [Rei, p.36, (1.10.4) & p.224, l.9].