The Rayleigh-Ritz method can be applied to arbitrary (nonorthogonal) basis functions [Mer2, p.139, l.-6-l.-2;
pp.146-149, §8.4].
The centrifugal potential Sℓ(ℓ+1)/2mr 2
in [Mer2, p.257, l.-20] refers to an inertial frame,
while the centrifugal potential energy -(2)m(Ω´r)2
in [Lan1, p.128, l.-5]refers
to a rotating frame.
[Wangs, p.12, (1-38)] shows how to make the definition of gradient
independent of a particular coordinate system.
In discussing wave speed in elastic media, what allows us to choose
different reference frames? For waves in a stretched spring, the velocity in
[Hall, p.297, Fig. 17-6] can go either right or left, but the absolute value of
the centripetal acceleration remains the same [Hall, p.297, l.-7].
How do we choose a convenient
reference frame? Since wave speed is ultimately determined by the characteristic
of the medium, we imagine that the wave pulse remains fixed in space and the spring
moves across the pulse with the same speed, but in the opposite direction. This allows
us to apply Newton's second
law to the medium and focus our discussion on the pulse length. A similar
analysis applies to [Hall, p.317, Fig. 18-2].
Remark. [Hall, p.297] proposes the above two philosophical questions, but Halliday's
answers are quite vague. Can philosophy only be studied after one masters
all the details in a subject? Should a beginner not be encouraged to try to
answer philosophical questions? In my opinion, when we encounter a philosophical
question worthy of contemplation, we should grab the opportunity and try
to answer the question. One should not ignore the opportunity
simply because the material is basic, and should not assume that we will have
another chance to think about it in a more advanced context. On the contrary, basic material is rich in
philosophy, while advanced material is often rich only in techniques.
Using vectors to eliminate the dependence of the system of coordinates and to show the direction of the wave propagation.
Example. [Matv, p.28, (2.34)] → [Matv, p.28, (2.35)].