The dynamics of particles and of rigid, elastic, and fluid bodies; Being lectures on mathematical physics

Arthur Gordon Webster

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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1904 Excerpt: ...different sets of coordinates. Let x', y', z' be its coordinates with respect to a set of axes fixed in space, and let x, y, z be its coordinates with respect to a set of axes moving in any manner. The position of the moving axes is defined by the position of their origin, whose coordinates referred to the fixed axes are, rl £, and by the nine direction cosines of one set of axes with respect to the other. Let these be given by the following table „ Since the axes Y, Z are perpendicular, their direction cosines satisfy the conditions, Hi) Ari + ftft + ftrs-o, and similarly, «ift + o,ft + «,/Js = 0. Thus the nine cosines are not independent, but, satisfying six conditions, may be expressed in terms of three parameters. These, with the three %, t, £, show the six degrees of freedom possessed by a rigid system. By interchanging the roles of the axes, and considering the direction cosines of X', Y', Z' with respect to X, Y, Z we find the equivalent conditions «1, + A2 + y1s = l, 112) «./ + /52 + JV = l, + +?-!, «!«, + ftft + yxyt = 0, 113) «2«s + ftft+ 72rs = 0 «3«i + ft ft + J'3ri = o. If we now differentiate the first of equations 109), supposing x, y, z to be constant, we obtain v-'d7x dt +y dt + rff + dt' m) » = dt =a; dt + » St' + df + dt'-at-x st + y dt + dt + dt' for the components in the directions of the fixed axes of the velocity of a point fixed to the moving axes. Let us now resolve the velocity in the direction which is at a given instant that of one of the moving axes. To resolve in the direction of the X-axis we have ( /, i b, drt, dt 115) vx = axi,' + Ojv/ + cv, =aL dt + % d + «,-ft + («. ft + « ft + «, ft) + »(« S + ft + S ft) + '(«.ft...

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