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].
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].
The same phenomenon viewed from different reference frames.
Example. [Cor, p.293, l.-2-p.294, l.2].
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].
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].
To unify Halmilton's principle, Fermat's principle and Maupertuis' principle of least action requires broad knowledge [Born, p.719, Appendix I].
These principles are theorems [Born, p.129, §3.3.2] rather than axioms.
[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].
Optics is a more refined subject than mechanics in the following sense:
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.
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].
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.
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:
(A fixed solid angle; one scatterer) A beam of identical particles passes through a central-field [Coh, p.906, (A-3)].
(A fixed angle of deflection; one scatterer) A beam of identical particles pass through a central-field [Lan1, p.49, (18.15)].
(A fixed angle of deflection; many scatterers) A beam of charged particles
is shot through a thin foil [Sym, p.138, (3.273)].
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).
(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
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
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,
[Rei, §1.5], [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:
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)].
Only a small region of integration contributes appreciably to the
integral [Rei, p.36, (1.10.4) & p.224, l.9].