Physical Interpretation in Differential Equations

Physical Interpretations in Differential Equations

Jacobi's necessary condition.
    The proof given in [Fomi, p.109, l.1-p.111, l.2] emphasizes its geometric meaning, while the proof given in [Akh, p.77, l.6-p.78, l.6] emphasizes its analytic meaning. Note the similarity between [Eis, p.161, Fig. 5-14] and [Fomi, p.110, Fig. 7].

Forces.
    To trace back to molecular forces [Lan7, p.4, l.7] and solve the problem using quantum mechanics is not always the best approach (Compare [Go2, p.103, (3-83)] with [Lan3, p.125, l.-9]). For deep understanding, it is helpful to have the second point of view. Any theorems in classical mechanics [Cou2, vol.2, pp605-607, §c] that are consistent with quantum mechanics, easy to visualize (intuitive and picturesque), and convenient to apply will be considered useful.

In order to give a physical interpretation of a differential equation, we must derive it from the physical point of view. However, in [Bir, pp.260-262, Chap. 10, §3. Physical interpretations], Birkhoff does nothing of this sort. All he does is summarize some physicists' results and then associate each differential equation with a physical phenomenon. He should refer reader to sources that provide good interpretations. For example, [Cou, vol. 1, pp. 244-252, chap. IV, §10.2 & §10.3; pp.286-308, chap. V, §3-§6].

The physical meaning of a concept when being taken out of context is very limited.
    The direct product [Jaco, vol. 2, p.200, l.3] and the Kronecker product [Jaco, vol. 2, p.208] of vector spaces are similar in the mathematical sense, but their physical meanings are entirely different. The former is related to the degrees of freedom [Coh, pp.153-154, §1; pp.160-163, §3], while the latter is related to the theory of group representation [Tin, pp.43-45, §representations]. The only benefit of taking the concept of the Kronecker product out of context is that enables us to focus on proving [Jaco, vol. 2, p.213, (16)] rigorously.

The idea of parallel transport in [Ken, p.32, l.10] can be formulated rigorously using differential geometry. See [O'N, p.322, Lemma 3.6].

A proof of a theorem does not necessarily help one understand the physical meaning of the theorem.
Example. The algebraic proof suggested by [Chr, part II, p.8, l.13] does not reveal the formula's [Chr, part II, p.8, Cor.3] physical meaning [Chr, part II, p.8, l.17-l.20].

(Calculus of variations) In [Wu, p.27, l.10], Wu says, "The dp's cannot be arbitrary." Indeed, when I first read Goldstein's Classical Mechanics [Go2, §2-3], I had the same worry as Wu. This was because Goldstein fails to use proper mathematical notations. However, after I read Gelfand's Calculus of variations, my worry gradually faded. Mr. T. Y. Wu is not only a great physicist but also a great essayist. I enjoyed reading many of his essays when I was in Taiwan. Wu is the only physicist who can gather evidence and know how to clearly formulate the question in his mind. Other physicists are either not honest enough to admit they have a problem or they are so confused that they do not even know how to phrase their questions in a complicated environment. Honesty and the courage to pursue truth are the keys to making progress in mathematics or physics. Now let us go back to Wu's question. When we say that p and q are independent variable, we are referring to the Hamiltonian function H(q, p, t) only. The reason is that this is the way the Hamiltonian is defined. p and q can be related. However, if they are, their relationship is outside the topic of the Hamiltonian. Therefore, we do not care about the relationship and still treat them as independent variables when we integrate or differentiate the Hamiltonian. In [Wu, p.27, l.7-l.8], Wu says, "… the condition for all varied trajectories to be traversed in the same t-t₀ precludes the dq·'s from being independent." Wu's worry is isolated, absorbed and hence eliminated by the last term (equal to 0 by [Wu, p.5, l.-3]) of the left-hand side of [Wu, p.7, (0-20)] through the device of integration by parts, so dp_k can be arbitrary (within the discussion of integration and differentiation of the Lagrangian).

Closure relation
    Suppose {u_i(r)} is a basis [Coh, p.99, l.-8; Ru2, p.90, Theorem 4.18] of the wave function space. Although the closure relation given in [Coh, p.100, (A-32)] is proved in the wave function space, the relation is also valid in any closed subspace of L²(Rⁿ) according to the argument given in [Coh, p.99, l.-4-p.100, l.6]. By [Bir, p.313, Theorem 11; Chou, p.166, l.-9-l.-8], [Mari, p.505, (12.160); Chou, p.166, (3.176) & (3.177)] are typical examples. Thus, if {u_i(r)} is an orthonormal base of a closed subspace of L²(Rⁿ), we may use the closure relation [Coh, p.100, (A-32)] to characterize the base in addition to the four equivalent ways listed in [Ru2, p.90, Theorem 4.18]. [Jack, p.67, l.15-p.68, l.-11] only provides a very rough discussion of closure relation. In [Jack, p.68, l.12-l.14], Jackson says, "Physicists generally leave the difficult job of proving completeness of a given set of orthonormal functions to the mathematicians." I do not think this is the proper attitude for a Berkeley physics professor or an exemplary motto for today's physics students to follow. Berkeley professors should not feel ashamed to ask experts questions. Every time the author of a textbook leaves a gap in his argument, he seriously impedes his readers' complete understanding. The hypothesis and conclusion of each statement in the textbook must be precise and absolutely clear. If the author leaves a gap in his argument, he should pinpoint the exact location where the readers can find resources to assist them to fill the gap. Sloppiness and irresponsibility should be removed from Berkeley physicists' future academic standard.

A bird's-eye view versus sectional views [Perr, p.22, l.5-p.24, l.14; p.25, l.7-l.-1]¬

The Fourier coefficient f_s of the force F(x,t) represents the component of F(x, t) effective in driving the normal coordinate s [Mari, p.523, l.8-l.9].

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