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Analogies in Mechanics



  1. The analogy between electric fields and magnetic fields
    1. [Sad, p.262, Table 7.1].
    2. Coulomb's law in integral form [Jack, p.26, (1.5)] [Jack, p.178, (5.14)].
    3.  
      1. Infinite line charge [Sad, p.127, (4.47)] Infinite line current [Sad, p.275, (7.20)].
      2. Infinite sheet of charge [Sad, p.128, (4.50)] Infinite sheet of current [Sad, p.276, (7.23)].
    4. [Jack, p.29, (1.14)] [Jack, p.179, (5.17)].
    5.  
      1. Differential form: [Jack, p.29, (1.13)] [Jack, p.179, (5.22)].
      2. Integral form: Gauss' law [Jack, p.28, (1.11)] Ampre's law [Jack, p.180, (5.24)].
    6. The analogy between the electric dipole moment and the magnetic dipole moment produced by a system of charges.
      1. definitions: [Lan2, p.96, (40.3)] [Lan2, p.104, (44.2)].
        Remark. For the physical interpretation of the magnetic moment, see [Jack, pp.186-187].
      2. potentials: [Sad, p.319, Table 8.2].
      3. far-fields: [Lan2, p.97, (40.8)] [Lan2, p.104, (44.4)].
    7. The analogy between electric and magnetic circuits [Sad, p.348, Table 8.4].
    Remark. The analogy provides a short cut to convince us that [Sad, p.285, (7.41)] is true. Please compare [Sad, p.285, l.9-l.12] with [Sad, p.285, l.-8-p.286, l.-1].

  2. Analogy is a natural way to raise the level of generalization.
    Example. Let us consider the right-hand side of the equality in [Lan2, p.100, l.-7]. Compare the first term with the second one.

  3. For the analogy between two theories, we would like to list their analogues as completely as possible. More importantly, we must search for the key idea behind the analogy. In other words, we must find a device (e.g., wave packets, the Lorentz transformation) that can establish a mathematical relationship between the equations of motion of the two theories. Through the key idea, we can easily understand, formulate, and deduce the rest of their analogues. We can also identify the cause that leads to the confusion (e.g., particles vs. waves; electric fields vs. magnetic fields).
        The key idea behind the analogy between
    1. geometrical optics and classical mechanics: [Lan2, p.130, l.-10-p.131, l.3].
    2. the wave vector of a wave and the momentum of a particle: [Lan2, p.131, l.12-l.24].
    3. electric fields and magnetic fields: [Lan2, pp.62-63, 24].
    4. Maupertuis's principle and Fermat's principle: the calculus of variations [Lan2, l.3-l.13].
    Remark. Special relativity searches for the mathematical foundation behind the analogy between electric fields and magnetic fields. Quantum mechanics searches for the mathematical foundation behind the analogy between particles and waves [Hec, p.36, l.c., l.-9-r.c., l.2]. It can be said that the development of each of these theories was initiated by the analogy and reached maturity with the successful explanation of the analogy.

  4. The limit of an analogy [Dit, pp.2-3. 1.4].

  5. The analogy between a LC circuit and a mass-spring system.
        The qualitative point of view (compare [Hall, p.625, Fig. 35-1] with [Hall, p.115, Fig.8-4]) the quantitative point of view (see [Hall, p.627, Table 35-1] and compare [Hall, p.628, (35-5) with [Hall, p.628, [14-6]).

  6. Suppose two theories A and B are similar. If a property is physical ly obvious in Theory A, but the corresponding property in Theory B may be difficult to recognize, then we may use the physical property of Theory A as a guide to establish the corresponding property of Theory B.
    Example [Hoo, p.103, l.-4-p.104, l.7]. Theory A = Lattice vibration waves. Theory B = Near free electron theory. Physical property = (The group velocity dw/dk vanishes at k=p/a) [Hoo, p.38, Fig. 2.4].

  7. How a free energy in thermodynamics behaves like a potential energy in mechanics.
    1. For a reversible process, work can be stored in the form of free energy and can be recovered completely [Reic, p.40, l.5-l.6].
    2. An equilibrium state is a state of minimum free energy [Reic, p.40, l.12-l.13].
      Proof. If the system does not do work in a process, then either the free energy does not change (reversible process) or it decreases (spontaneous process).

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