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As a new varied path Ct. Taking into account that time instants t and t + dt correspond to the positions Mt and MI*, respectively, and assuming that the position vectors of the points of the varied path are differentiable yields Mt MI* = d (ri + bri) = Now by virtue of the evident equality we have dri + dbri. 38 1. Basic definitions or d8ri - 8dri = 0 (i = 1, ... , N). 5) Then due to (4) we obtain (i = 1, ... , N). 1) for Cartesian coordinates we have aev (d8qs La s=1 qs n 8dqs) = 0 (v = 1, ...

By means of the Kronecker delta one can cast it in the form •• lk . 1m = >: Ukm 3 3 = { 01 (k (k # m), = m). 3) Hence i~ . 4) askatk = Dst· k=l The latter yields six relations linking the nine direction cosines of Table 1. These relations are given by ail + aI2 +aI3 = 1, a§1 = 1, = 1, + a§2 + a§3 a~l + a~2 + a~3 + a12a22 + a13a23 = 0, a21 a31 + a22a32 + a23a33 = 0, a31 all + a32a12 + a33a13 = O. 5) The vector product i1 x i2 is defined as a unit vector perpendicular to the plane containing i1 and i2 and points in the direction of axis Oz, in other words i1 x i2 = i 3.

When it is desirable to avoid primes we use the notation O~'TJ( (or O~ 1 ~2~3) for inertial axes and Oxyz (or OXIX2X3) for moving axes. The position of the system of moving axes Ox' y' z' is given by the coordinates of its origin Xo, Yo, zo (or -=--+ its position vector TO = 00) and the table of direction cosines II II II II x' y' z' II II II II x y au a12 a21 a22 z a13 a23 a31 a32 a33 II II II II Table 1 of direction cosines A. I. Lurie, Analytical Mechanics © Springer-Verlag Berlin Heidelberg 2002 48 2.

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