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Solutions in Mechanics

  1. By using the Hamilton-Jacobi theory, we can directly obtain the frequency of periodic motion without finding a complete solution to the equation of motion [Go2, p.461, l.-15] (Example: Linear harmonic oscillator [Go2, p.462]).

  2. Qualitative approach vs. quantitative approach.
        The qualitative approach and quantitative approach complement each other. One is neither inferior to the other nor less informative than the other. They have different emphases. The qualitative approach emphasizes physical meanings, geometric features, and the big picture. It provides a guideline for the analytic [Eis, H-1, l.2] and numerical approach (e.g., it helps save effort in numerical calculation [Eis, G-5, l.26-l.36]). In contrast, the quantitative approach emphasizes precision (e.g., qualitative ; [Eis, p.246, l.8-l.9]; quantitative [Eis, p.246, l.10]: [Eis, p.246, (7-29)] ) and details (e.g., [Coh, p.73, (29)] gives a proof of [Eis, p.201, (6-51)]).

  3. Consideration of physical examples often makes the solving process more effective.

    Example 1. Compare [Hec, p.188, r.c., l.-13-l.-8] with [Hec, p.188, l.c., l.-5-r.c., l.14].

    Example 2. (Method of image)
        The simple consideration (of the surface-charge density on a grounded sphere induced by a point charge) helps us find a solution without going through the formal process for solving a P.D.E. [Jack, p.58, (2.2) & p.59, (2.4)].

  4. The more in depth we study a problem, the more precisely we can pinpoint the answer.
    Example 1. Compare [Coh, p.1161, Fig.4, l.2-l.3] with [Eis, p.447, Fig. 13-2, l.3-l.5].
    Example 2. Compare [Hoo, p.60, (2.54)] with [Eis, p.407, l.8].

  5. Various methods of solving the Poisson equation are given throughout [Wangs, pp.171-199, Chap. 11]. It is equally important to classify these solutions in terms of the precise language of differential equations. The word "Systematic" used in [Wangs, p.185, l.14] is neither rigorous nor specific enough for a proper classification.
        [Wangs, p.69, (5-7)] is a particular solution of the inhomogeneous equation [Wangs, p.171, (11-1)]. This solution is obtained physically by considering the sources [Wangs, p.37, (1-148) & l.-1]. The solutions obtained by the method of separation of variables in [Wangs, pp.185-198, 11-4) & 11-5] are the general solutions of a homogeneous equation.
    Remark. In [Wangs, pp.185-198, 11-4) & 11-5], we use separation of variables to solve the Laplace equation, while in [Lan1, pp.149-153, 48] we use separation of variables to solve the Hamilton-Jacobi equation. They are closely related because the Hamiltonian in [Lan1, p.151, l.7] contains the Laplacian term.

  6. (Poisson's equation)
        In essence, the proof of [Kara, p.100, (5.75)] and that of [Wangs, p.72, (5-15)] are the same. The proof of [Kara, p.100, (5.75)] emphasizes the properties of the d-function without mentioning solid angles, while the proof of [Wangs, p.60, (4-10)] emphasizes solid angles without mentioning the concept of the d-function. The concept of solid angle is good for physical interpretations, while the d-function is good for generalization of the method of solving PDE's. The limiting procedure in [Jack, p.35, l.5] is an artificial mathematical construct which helps visualize the d-function [Coh, p.1470, (7)-(11)], but is not good for describing the properties of the d-function. [Kara, p.99, (5.71)] is an exact formula rather an approximation. Thus, the second-order approximation in [Jack, p.35, l.-10] does not have more advantages than the first-order approximation. The unnecessarily complicated calculations in proving [Jack, p.34, (1-28)] by substituting [Jack, p.35, (1.17)] can be obviated by the more physical approach in the proof of [Wangs, p.60, (4-10)].
    Remark. [Jack, p.35, (1.17)] has the form of [Ru3, p.192, (3)].

  7. We solve problems through patterns. For example, after reading [Arf, pp.521-525, Example 8.7.2], we will be able to solve [Ashc, p.352, Problem 3(b)] because the argument of their solutions has the same pattern. The argument of the former is more complicated than that of the latter. The complication lies in the technique of calculating contour integrals. As another example, for the proof of the fact that the force on each atom in a solid vanishes, the case of a simple cubic lattice is similar to the case of the hexagonal close-packed lattice. The argument of the latter case is complicated by the requirement of considering more symmetries [Pei, p.7, l.-4-p.8, l.10].

  8. Substituting a given function into an equation is not an effective way to construct a solution. An effective construction usually comes from physical considerations. See [Ashc, p.576, footnote 14].

  9. Solving the Maxwell equations in macroscopic media.
    1. Suppose the sources are given. Theoretically, the two equations in [Jack, p.2, (I.1b)] should provide the solutions for the electric and magnetic fields.
    2. The exact solutions are impossible: The number of individual sources is prohibitively large and the details of field variations over atomic distances is irrelevant [Jack, p.13, l.20-24].
    3. Only the average of a field over a large volume is relevant. The macroscopic Maxwell equations still preserve the original form [Jack, p.13, l.-19-l.-13].
    4. The electric displacement and magnetic field: [Jack, p.13, (I.9)] (the bound charges and currents appear in the equations via P, M, and Qab').
    5. Constitutive relations
      1. Weak fields: (using Fourier transforms) [Jack, p.15, (I.11)].
      2. Strong fields: [Jack, p.18, (I.12)].
    6. Solving the Maxwell equations with boundary conditions.
      1. The strategy: [Wangs, p.133, l.1-l.3; Fig. 9-1 & Fig. 9-2].
      2. The formulas: [Wangs, p.134, (9-7); p.136, (9-18)].
        Remark. W 0 as h 0 [Wangs, p.134, l.11] because F is bounded in the small cylinder. In [Wangs, p.134, l.3], Wangsness claims that W ~ h [Wangs, p.2, l.12-l.14]. This statement of Wangsness is incorrect. Wangsness commits the same mistake again in [Wangs, p.135, l.-14].
      3. (Summary) The method of calculating the fields across the surface of discontinuity: [Wangs, p.136, l.13-l.21].
      Remark. The above discussion is restricted to the study of discontinuity across the boundary. For a more general discussion of the Poisson equation with boundary conditions, see [Jack, 1.8, 1.9].

  10. Our approach to solving a physics problem is to tap more resources and then find the most effective method.
        [Chou, p.236, l.-8-p.237, l.9] proves an = bn = gn = dn = 0 (n 2) without using the top two (/q)-equations of [Jack, (5.119)]. As a consequence, the calculations in the proof are unnecessarily complicated. In fact, in order to prove an = bn = gn = dn = 0 (n 2), all we have to do is compare corresponding coefficients of Pn'(cos q) by using the top two (/q)-equations of [Jack, (5.119)] and the linear independence of Pn'(cos q) [Guo, p.219, (7)].
    Remark. [Chou, (5.138)] is simpler than [Chou, (5.137)], so Choudhury tries to solve [Chou, (5.138)] first. This approach makes his method of solving the system of linear equations [Chou, (5.137) & (5.138)] very effective.

  11. Damped and forced harmonic oscillators [Sym, 2.3, 2.7-2.11; Edw, 3.4-3.7; 9.4]
        On the one hand these two devices provide a primitive model about how to effectively solve the general type of ordinary differential equations. On the other hand they give physical interpretations for the solutions. Unfortunately, [Sym, 2.3, 2.7-2.11] provide only the outline of both mathematical solutions and their physical interpretations. To appreciate the importance of [Sym, 2.11] requires a good knowledge of the theory of ordinary differential equations. To gain complete understanding of the physical  meaning of [Sym, 2.3, 2.7-2.11] one must have a good knowledge of electromagnetism [Wangs, 25-7, 27-1 & 27-2].
    1. The switch in [Wangs, Fig 27-2] is an important element for the derivation of [Wangs, (27-9)]. The omission of the switch in [Chou, Fig. 6.12] shows that the material is outdated.
    2. Wangsness uses Kirchhoff's law [Wangs, (27-6)] and Lenz' law to derive [Wangs, (27-7)]. Because Choudhury does not use the above two laws, the sign determination of voltages in [Chou, (6.74)] is not clear.
    3.  
      1. (Steady-state response)
            The force term in [Sym, (2.148)] is the typical term of a periodic force [Sym, (2.205)].
      2. (Green's function)
            Physically, a d-function [Sym, (2.191)-(2.193)] can be interpreted as an impulse [Sym, p.57, l.8-l.9]. Green's function [Sym, (2.190) or (2.211)] is derived from the transient solution [Sym, (2.133) or (2.187)]. The integral in [Sym, (2.212)] can be interpreted as the right-hand side of [Sym, (209)] because a force can be considered a sum of impulses [Sym, p.61, Fig. 2.8].
        Remark. It is worth noting that Green's original approach [Sym, p.61, l.4] clarifies the intuitive meaning of Green's function, while both Birkhoff-Rota's [Bir, 2.8; p.75] and Jackson's [Jack, pp.38-40] approaches obscure the intuitive meaning of Green's function.
    4. How an old crystal radio works [Edw, p.226, l.12-l.-10].

  12. Green functions for the wave equation
    1. The wave equation is defined by [Jack, (6.32)]. The only difference between the wave equation and Poisson's equation in electrostatics is that the formal equation has an extra term -c-2(2Y/t2). This extra term is caused by the fact that the propagation of electromagnetic disturbances requires time. Therefore, the method for solving the wave equation is similar to that for solving Poisson's equation except we have to replace current time by retarded time [Jack, (6.37)].
    2. By means of a Fourier transform with respect to frequency [Jack, (6.33) & (6.34)] we remove the explicit time dependence and obtain the inhomogeneous Holmholtz wave equation [Jack, (6.35)]. The Green function appropriate to [Jack, (6.35)] satisfies [Jack, (6.36)].
    3. [Jack, (6.36)] [Jack, (6.37)]. The solutions of [Jack, (6.37)] are [Jack, (6.40)].
    4. In [Jack, (6.40)], Gk(+)(R) represents a diverging spherical wave propagating from the origin [Wangs, p.379, l.8], while Gk(-)(R) represents a converging spherical wave.
    5. The solutions of [Jack, (6.41)] are [Jack, (6.43)].
      Proof. (2+k2) G(x,w,x',t') = -4pd(x-x')eikR  (the proof is similar to that of [Jack, (6.35)]).
      By [Jack, (6.40)], G(x,w,x',t') = (eikR/R)eiwt.
      By [Jack, (6.33)], we have [Jack, (6.42)] ( [Jack, (6.43)]).
    6. By [Ru3, p.193, (4)], the solutions of [Jack, (6.32)] are the integrals in [Jack, p.245, l.13].
    Remark. The proof in [Chou, 7.4] using the residue theorem [Ru2, p.241, Theorem 10.42] keeps the discussion of the d-function as a whole. In contrast, Jackson discusses a special case [Jack, p.244, l.1-l.2] first, and then provides a remedy [Jack, p.244, (6.38)] for the portion of the domain of the d-function that the special case does not cover. One may wonder if Jackson puts the pieces back together in a seamless manner after breaking his argument  into separate discussions. For example, one may question whether Jackson neglects to check the compatibility of his argument for the parts of cases that overlap. [Mer, p.291, l.16-p.293, l.8] and [Arf, p.520, l.1-p.525, l.5] offer better ways to make Jackson's proof rigorous. From [Arf, p.523, (8.212)], we may recognize that [Arf, p.524, (8.128)] is the Green function of [Arf, p.522, (8.205b)]. The proof given in [Coh, p.1478, l.-5-p.1479, l.8] is simpler than the above three. However, the proof given in [Born, 2.1.2] is derived from the most natural physical motive.

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