and Hamilton's equation (5) of last section becomes where is a self-conjugate linear and vector function, whose constituent vectors are fixed in the body in its initial position. Then the previous equation may be written 384.] It is easy to see what the new vectors 7 and represent. For we may write (2) in the form (2) from which it is obvious that n is that vector in the initial position of the body which, at time t, becomes the instantaneous axis in the moving body. When no forces act, y is constant, and is the initial position of the vector which, at time t, is perpendicular to the invariable plane. 385.] The complete solution of the problem is contained in equations (2), (3) above, and (4) of § 356 *. Writing them again, we have gn= 24, vq = q5 φη = ζ. We have only to eliminate and 7, and we get η, (4) (2) (3) (5) in which q is now the only unknown; y, if variable, being supposed known in terms of q and t. It is hardly conceivable that any simpler, or more easily interpretable, equation for q can be presented *To these it is unnecessary to add Tq = constant, as this constancy of Tq is proved by the form of (4). For, had Tq been variable, there must have been a quaternion in the place of the vector ŋ. In fact, until symbols are devised far more comprehensive in their meaning than any we yet have. 386.] Before entering into considerations as to the integration of this equation, we may investigate some other consequences of the group of equations in § 385. Thus, for instance, differentiating (2), we have which gives, in the case when no forces act, the forms -1 (6) (7) To each of these the term q1yq, or q1yq, must be added on the right, if forces act. 387.] It is now desirable to examine the formation of the func-. tion 4. By its definition (1) we have φρ =2.m (a Sap-a2p), Hence =-Σ.ma Vap. -Spop.m (TVap)2, so that -Spop is the moment of inertia of the body about the vector p, multiplied by the square of the tensor of p. Thus the equation Spopp-h2, evidently belongs to an ellipsoid, of which the radii-vectores are inversely as the square roots of the moments of inertia about them; so that, if i, j, k be taken as unit-vectors in the directions of its axes respectively, we have A, B, C, being the principal moments of inertia. Consequently op={Ai - { Ai Sip + BjSjp + Ck Skp}.. Thus the equation (7) for ŋ breaks up, if we put η n = iw1 + jw2 + kwz, into the three following scalar equations Aw1+ (C—B) w2W3 = 0, (8) (9) which are the same as those of Euler. Only, it is to be understood that the equations just written are not primarily to be considered as equations of rotation. They rather express, with reference to fixed axes in the initial position of the body, the motion of the extremity, w1, w2, w, of the vector corresponding to the instantaneous axis in the moving body. If, however, we consider w1, wą, w as standing for their values in terms of w, x, y, z (§ 391 below), or any other coordinates employed to refer the body to fixed axes, they are the equations of motion. Similar remarks apply to the equation which determines (, for if we put $ = iw1 + jw2+kw3, 2 (6) may be reduced to three scalar equations of the form 388.] Euler's equations in their usual form are easily deduced from what precedes. For, let whatever be p; that is, let represent with reference to the moving principal axes what represents with reference to the principal axes in the initial position of the body, and we have But perhaps the simplest mode of obtaining this equation is to start with Hamilton's unintegrated equation, which for the case of no forces is simply so that #=Vew+Vεw = we2 ― €Sew + Vew, Σ.m (Vew Sew-éw2 + w Sew) = 0. If we look at equation (1), and remember that simply in having differs from substituted for a, we see that this may be the equation before obtained. The first mode of arriving at it has been given because it leads to an interesting set of transformations, for which reason we append other two. But, by the beginning of this section, and by (5) of § 382, this is again the equation lately proved. Perhaps, however, the following is neater. It occurs in Hamilton's Elements. 389.] However they are obtained, such equations as those of §387 were shewn long ago by Euler to be integrable as follows. with other two equations of the same form. Hence so that t is known in terms of s by an elliptic integral. Thus, finally, nor may be expressed in terms of t; and in some of the succeeding investigations for q we shall suppose this to have been done. It is with this integration, or an equivalent one, that most writers on the farther development of the subject have commenced their investigations. 390.] By § 381, y is evidently the vector moment of momentum of the rigid body; and the kinetic energy is so that we have, for the equations of the cones described in the η initial position of the body by ŋ and ¿, that is, for the cones described in the moving body by the instantaneous axis and by the perpendicular to the invariable plane, h2¿2 + y2 85 p¬18 = 0, h2 (pn)2 + y2Snon = 0. This is on the supposition that y and I are constants. If forces act, these quantities are functions of t, and the equations of the cones then described in the body must be found by eliminating t between the respective equations. The final results to which such a process will lead must, of course, depend entirely upon the way in which t is involved in these equations, and therefore no general statement. on the subject can be made. 391.] Recurring to our equations for the determination of 2, and taking first the case of no forces, we see that, if we assume ŋ to have been found (as in § 389) by means of elliptic integrals, we have to solve the equation qn = 2q*, that is, we have to integrate a system of four other differential equations harder than the first. Putting, as in § 387, n = iw1 + jw2 + kwz, where w1, w, w are supposed to be known functions of t, and *To get an idea of the nature of this equation, let us integrate it on the supposition that n is a constant vector. By differentiation and substitution, we get And the interpretation of q()q-1 will obviously then be a rotation about ʼn through the angle tTn, together with any other arbitrary rotation whatever. Thus any position whatever may be taken as the initial one of the body, and Q() Q1 brings it to its required position at time t=0. -1 |