By J. L. Synge (auth.), S. Flügge (eds.)
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2 The trace (or spur) -of a matrix is the sum of the diagonal elements. 3 In Sect. 1), confusion may be avoided by changing to (I,], K) or (e1 , e2 , e3). 4 It will become clear in Sect. 15 that the transformation is proper. For a proof, see MuRNAGHAN (op. cit. in Sect. 10, p. 298), where the group theory connected with the PAULI matrices is discussed. v-1 J. L. 24 SYNGE: Classical Dynamics. Sect. 15. x. 2) . Consider the transformation P' = U 3 (X) P U~ (x) = (c 1 - is 6 3) (x 6 1 + y6 +z6 2 3) (c 1 + is 6 3).
9) where m is the total mass of the system. This formula is particularly useful when 0* is the mass-centre. 11) h 0 may be called the orbital angular momentum and h* the spin angular momentum, to borrow the terms of quantum mechanics. 11), or the absolute velocities. It is well to emphasise that, in any computation of angular momentum, we need to specify (i) a frame of reference relative to which velocities are measured, and (ii) a point about which moments are taken. : m;r;(w · r;). 13) where (w1 , w2 , w3) are the components of wand A, B, C,F, G, Hare the moments and products of inertia relative to the triad.
Consider a rigid lamina which can slide freely over a fixed plane; this is a scleronomic holonomic system with three degrees of freedom. But now suppose that a small sharp blade is fixed in the lamina, the blade being capable of motion only in the direction of its length. If (x, y) are the Cartesian coordinates of the blade and {} its inclination to the x-axis, then (x, y, {}) form a system of generalized coordinates for the lamina, but they are subject to the non-integrable relation {~=tan-D. 1} The number of generalized coordinates cannot be reduced below three, but these three coordinates are not freely variable.