with two similar formulae.
| Fig. 70. |
Again, if the instantaneous position of G be taken as base, the angular momentum of the absolute motion is the same as the angular momentum of the motion relative to G. For the velocity of a particle m at P may be replaced by two components one of which (v) is identical in magnitude and direction with the velocity of G, whilst the other (v) is the velocity relative to G. The aggregate of the components mv of momentum is equivalent to a single localized vector Σ(m)·v in a line through G, and has therefore zero moment about any axis through G; hence in taking moments about such an axis we need only regard the velocities relative to G. In symbols, we have
| Σ { m(yż − zẏ) } = Σ(m)· ( y | dz | − z | dy | ) + Σ { m (ηζ − ζ̇η̇) }. |
| dt | dt |
(5)
| Fig. 71. |
since Σ(mξ) = 0, Σ(mξ̇) = 0, and so on, the notation being as in § 11. This expresses that the moment of momentum about any fixed axis (e.g. Ox) is equal to the moment of momentum of the motion relative to G about a parallel axis through G, together with the moment of momentum of the whole mass supposed concentrated at G and moving with this point. If in (5) we make O coincide with the instantaneous position of G, we have x, z, z = 0, and the theorem follows.
Finally, the rates of change of the components of the angular momentum of the motion relative to G referred to G as a moving base, are equal to the rates of change of the corresponding components of angular momentum relative to a fixed base coincident with the instantaneous position of G. For let G′ be a consecutive position of G. At the instant t + δt the momenta of the system are equivalent to a linear momentum represented by a localized vector Σ(m)·(v + δv) in a line through G′ tangential to the path of G′, together with a certain angular momentum. Now the moment of this localized vector with respect to any axis through G is zero, to the first order of δt, since the perpendicular distance of G from the tangent line at G′ is of the order (δt)2. Analytically we have from (5),
| d | Σ { m (yż − zẏ) } = Σ(m)· ( y | dz2 | − z | d2y | ) + | d | Σ { m(ηζ − ζη̇) } |
| dt | dt2 | dt2 | dt |