Reductions in Mechanics

  1. If the new theory is parallel to the old theory, we may carry the results of calculations in the old theory over to the new one. It is unnecessary to start the new theory from scratch.
        The results of calculations in the active case can be carried over to the passive case (Compare [Mer2, p.76, l.10] with [Mer2, p.77, l.-11]; compare [Mer2, p.76, l.16] with [Mer2, p.77, (4.118)]). Thus, it is unnecessary to repeat similar calculations for the passive case. The method for defining the boosted wave functions and potentials for the active case [Mer2, p.76, (4.112)] can be used to define their counterparts for the passive case [Mer2, p.77, (4.116)].

  2. The general formula of the reaction force and that of the reaction moment in [Lan7, p.43, (12.6) & (12.7)] are very complicated, but for the special cases where the plate is clamped or supported [Lan7, p.43, Fig.4] the formulas reduce to one or two terms [Lan7, p.44, (12.9), (12.10) & (12.11)]. This is because some terms in the general formula vanish and some terms cancel each other. In these cases, it is important to give a simple explanation [Lan7, p.44, l.1-l.2; l.16] of why these terms vanish rather than to give detailed calculations.

  3. A two-term recursion relation [Lev2, p.136, l.-14-l.-10].

  4. Model reductions are the crucial steps to understanding the hydrogen atom [Coh, p.775, l.1-l.20].
    1. Stationary states of a particle in a central potential [Coh, p.776, l.1].
    2. Reduction of the two-noninteracting-particle problem to two separate one-particle problems using separation of variables [Lev2, p.126, l.11].
    3. If the potential energy of interaction is a function of the relative coordinates alone, the two-interacting-particle problem can be reduced to two separate one particle problems [Lev2, p.127, l.1].
    4. Confinement of our study in the center of mass frame [Coh, p.791, l.2].

  5. Choose the Lorentz condition to decouple the wave equations containing both the electric and magnetic fields [Sad, p.388, l.-6].

  6. In order to avoid complicated calculations and to obtain the desired solution directly, we may switch the roles in a previous result with proper modifications.
    Example. [Wangs, p.502, l.9-l.12].

  7. How we reduce calculations when deriving a formula.
    1. Using general properties instead of special ones.
      Example. The derivation of [Lan3, p.67, (26.5)] is simple because Landau uses the general property [Lan3, p.67, l.9].
      In contrast, the derivation of [Jack, p.557, (11.143)] is involved because Jackson uses the special property [Lan3, p.557, l.8].
    2. Reducing to a simpler model so that we can avoid repeating the same calculations used in the simpler model.
      Example. When Landau derives [Lan3, p.46, (16.8)], he reduces his calculations from the case of a particle moving in an electromagnetic field to the case of a free particle. Therefore, his derivation is simple. In contrast, when Jackson derives [Jack, p.582, (12.17)], he substitutes [Jack, p.582, (12.16)] into [Jack, p.582, (12.12) & (12.15)] with brutal force. Therefore, Jackson's derivation is less methodical and more complicated.