Coordinate Systems in Mechanics

  1. The Rayleigh-Ritz method can be applied to arbitrary (nonorthogonal) basis functions [Mer2, p.139, l.-6-l.-2; pp.146-149, 8.4].

  2. The centrifugal potential S(+1)/2mr 2 in [Mer2, p.257, l.-20] refers to an inertial frame, while the centrifugal potential energy -(2)m(Ωr)2 in [Lan1, p.128, l.-5]refers to a rotating frame.

  3. [Wangs, p.12, (1-38)] shows how to make the definition of gradient independent of a particular coordinate system.

  4. 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.

  5. 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)].