Generalization in Mechanics

  1. Generalization.

    1. It is enough to prove [Eis, p.M-1, (M-2)] for the case of functions of a single variable [Eis, p.M-1,l.- 11] because the linear combinations of products of functions of a single variable form a dense subset of the function space. Thus we have a more effective argument (It is easy to calculate dr/dx, see [Eis, p.M-1, l.- 1].) in proving [Eis, p.M-1, (M-2)] for the case of spherical coordinates than the general case [Cor, p.23, (1-83)].

    2. After a formula [Faraday’s law: Jack, p.210. (5.139)] is established in the general case, it is adequate to determine its constant [Jack, p.211, (5,141)] by restricting ourselves to a special case [Jack, p.211, l.−12].
      Example. The derivation from [Jack, p.209, (5.135)] to [Jack, p.210. (5.139)] is valid for the general case (no approximation is involved). We restrict ourselves to the special case of Galilean invariance only for the evaluation of constant k.

    3. [Jack, p.40, l.17-l.31] seems to try to tell us something important, but fails to deliver a clear theme. 
          Although it is difficult to go from the general case to a special case, it is always interesting to see how we use the flexibility of the general solution [Jack, p.40, l.17] to find the most effective approach [The method of images: Jack, p.40, l.26] to the solution for a given specific case.

    4. Choosing the right tools for generalization.
          The coordinate independent form of vector analysis is not the right tool to generalize the invariance under rotation to the time-space domain because time and space are dependent [Rob, p.81, l.−12]. We must choose the natural tool of tensor algebra to generalize the geometric operation (rotation) to 4-dimentional space [Rob, p.81, l.−4].

    5. Occasions that require specifications.
          When we derive one mathematical expression [Lan2, p.28, (9.18)] from another one [Lan2, p.28, (9.17)], we would like to specify the expressions by endowing them with physical meanings [Rob, p.84, l.−12-l.−2] so that we will be able to understand intuitively the incentive and direction of derivation. In the above example, a brutal force calculation is also a possible alternative, but the derivation would be meaningless.

    6. Although both [Go2, p.7, l.12] and [Lan1, p.19, l.−8] are called Conservation theorem for angular momentum, we must point out two differences between these two proofs:
      1. Linear momentum is defined by the Lagrangian in [Lan1, p.19, l.8], while linear momentum is mv in [Go2, p.6, l.−6]. Thus the linear momentum in [Go2, p.6, l.−6] is a special case of [Lan1, p.19, l.8].
      2. Goldstein's proof requires the use of the strong law of action and reaction [Go2, p.7, l.6], while Landau's proof removes this restriction. Thus interaction is irrelevant to the conservation of angular momentum [Lan1, p.19, l.−6].

    7. The same argument can be used to derive both the constant velocity for a free particle [Lan2, p.27, (9.10) ] and the equation of motion for the general case using the principle of least action. The special case provides a motive [Lan1, p.142, l.11-l.13] for the argument. If we replace the free particle with the general setting, the same argument will produce the right form (Lagrange's equations [Lan1, pp.2-3, §2]) of the equation of motion for the general case.

    8. Conservation of areal velocity.
      1. Scope.
        Special case: inverse square law of force [Gol, p.79, (3.53)].
        General case: central force [Gol, p.61, l.10-l.13].
      2. Purpose.
        Broadening our view and raising our understanding to a higher level- Conservation of angular momentum [Gol, p.61, l.3-l.5].

    9. Lagrange's equations.
      1. Why the general form is necessary [Gol, p.21, l.8-l.10].
        Special form: Conservative systems [Gol, p.18, (1-50)].
        General form [Gol, p.19, (1-54)].
      2. Advantage of generalization. Simplification [Gol, p.22, l.−5].
      3. Agreement: Linear momentum [Gol, p.24, l.3]; Angular momentum [Gol, p.25, l.9-l.10].

    10. Scattering angle of a particle by a fixed center of force [Gol, p.82, Fig. 3-13]
      → Scattering angle by the two-body central force [Gol, p.87, (3-72)].
      (For agreement, see [Gol, p.87, l.−3]).

    11. Distinguishing central from peripheral after generalization.
          In order to rigorously define the derivatives of the Dirac measure [Ru3, p.141, l.−4] (or δ-function [Coh, p.1468]), we have to define its attached shadows as well [Ru3, p.136, (2) & p.143, Definition 6.12]. For application, we must not be distracted by these shadows because we never use them (see [Coh, p.1476, (54)]).

    12. The main feature of generalization. In regard to reducing the dimension of integration, Green's theorem [Kap, p.295], (Kap, p.227, (4.11)), Stokes' theorem and Gauss' theorem [Kap, p.298, l.11] are all related to one another.

    13. We may obtain a specific formula [Cor, p.24, (1-84)] from the general formula in [Cor, p.23, (1-83)] simply by substituting data [Cor, p.19, Table 1-1].

    14. In practice, a specific method is usually more solid and precise than a general method because the validity of the former method is based on delicate analysis and strong evidence. Although a general method is crude, it can be applied to more widely [Rei, p.424, l.−7-p.428, l.3].

    15. In view of the procedure of a generalization, the relationship between the motivation and the benefits of the generalization may not be straightforward.
      Example. Covariant Lagrangian formulations.
      1. Motivation of the generalization: Covariant formulation [Go2, p.326, l.20].
      2. Method of the generalization: Choose θ [Go2, p.327, l.7].
      3. Determination of which properties we should preserve [Go2, p.327, (7-159); p.329, l.−2; p.330, l.11] and which properties we need not preserve [Go2, p.329, l.−3].
      4. Benefits of the generation: [Go2, p.329, l.8; p.330, l.−1].

    16. The higher order approximation is not just for theoretic interests; there are practical examples that actually demand higher order precision [Go2, p.519, l.−13].

    17. During the procedure of generalization, on the one hand, one should recognize the general pattern [Go2, p.522, l.−5-p.523, l.9]; on the other hand, one should distinguish the creative and essential part [Go2, p.521, l.−9] from the routine part.

    18. Only through generalization may we reveal the following subtle fact:
      The property that E and H are transverse and orthogonal [Cor, p.518, l.7] is shared by all kinds of electromagnetic plane waves [Born, p.23, (4)], not just the harmonic electromagnetic plane wave [Cor, p.518, (28-7) & (28-8)].
      Remark. For a given property P, generalization serves to seek the weakest hypothesis that still implies property P.

    19. In expository writing, one must clearly indicate whether one is considering the general case or one specific case.
          Because Cohen-Tannoudji fails to indicate his stance in [Coh, p.781, l- 6-l.- 4], his readers might think that he is referring to a specific case, when in fact he is trying to describe a general case. In contrast, [Lan3, p.105, l.11-l.13] clearly indicates that Landau refers to the general case, and explains why the specific case in [Lan3, p.105, l.12-l.13] can be considered typical (i.e. the general case).

    20. Generalization helps us narrow down the essential causes of nondegeneracy, and also helps us reduce the calculations involved at the same time. We may use the following three methods to prove the nondegeneracy of eigenvalues of a linear harmonic oscillator:
      1. Find all the solutions directly [Mer2, p.83, (5.21) & (5.22)].
        Remark. This method requires that we solve the Schrödinger equation for every value of n.
      2. Establish a ladder relationship on nondegeneracy between consecutive eigenvalues [Coh, pp.494-495].
        Remark. In order to prove the nondegeneracy, we only need to solve the Schrödinger equation for the ground state.
      3. Consider the constancy of the Wronskian [Mer2, p.83, l.-10].
        Remark. In order to prove the nondegeneracy, it is unnecessary to solve any of the Schrödinger equations.

    21. The general case helps us identify the fundamental reason for a consequence.
          In [Sad, p.134, l.-13-l.-10], Sadiku gives a reason why VAB is independent of the path taken. However, the reason Sadiku provides is only for a special case; the reasons for other cases cannot be formulated in the same way. The fundamental reason (simply connectedness) for the general case is given by [Cou2, vol.2, p.104, l.16-l.21].

    22. A general case vs. its special cases.
          [Cor, §5.1] discusses the electric dipole; [Fan, §3.5] discusses the point dipole; [Cor, §5.2] discusses the linear electric quadrupole. Their various views give one a solid feeling about the meaning of the dipole or quadruple because their arguments do not distract one's attention to generalization. [Wangs, (8-7) & (8-8)] give the multipole expansion of the electric potential. This expansion is the general case. If we do not study the general case (more specifically, the dipole term of the multipole expansion), we will not be able to understand why we have to study the point dipole [Wangs, p.119, l.-11-p.120, l.2]. If we do not study the general case, we might think the potential of the design in [Cor, §5.2] accidentally coincides with the quadrupole term of the multipole expansion. In fact, the design is calculated to eliminate the monopole term and the dipole term [Wangs, p.119, l.3] so that the remaining dominate term is the quadrupole term.

    23. Links {1, 2, 3}.

  2. Degeneration.

    1. Generalization « degeneralization.
      1. [Sad, p.117, (4.33)] (for a volume charge) « [Sad, p.117, (4.34)] (for a point charge).
      2. [Sad, p.118, Example 4.4(a)] (for a ring) « [Sad, p.119, Example 4.4(c)] (for a point charge).
      3. A special case [Sad, p.209, l.-3; p.210, l.-5] ® the general case [Sad, p.210, l.-4] (by solving Laplace's equation with boundary conditions) ® the special case [Sad, p.210, l.-1] (check the consistency by substitution).
      4. [Sad, p.239, l.9-l.10].