The Dynamics of Particles and of Rigid, Elastic, and Fluid Bodies; Being Lectures on Mathematical Physics (Paperback)

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...