precesses about the direction of g (i.e. the vertical) with a mean angular velocity Spr (ul/M)g cos a (2) which is small compared with Senu Spr in no a FIG. 50 In this approximation the quantities M and cos a in formulae (1) and (2) are constants, although they are not exact integrals of the motion. To the same accuracy they are related to the strictly conserved quantities E and M3 by M3 = M cos a, §36. Euler’s equations The equations of motion given in §34 relate to the fixed system of co- ordinates: the derivatives dP/dt and dM/dt in equations (34.1) and (34.3) are the rates of change of the vectors P and M with respect to that system. The simplest relation between the components of the rotational angular momentum M of a rigid body and the components of the angular velocity occurs, however, in the moving system of co-ordinates whose axes are the principal axes of inertia. In order to use this relation, we must first transform the equations of motion to the moving co-ordinates X1, X2, X3. Let dA/dt be the rate of change of any vector A with respect to the fixed system of co-ordinates. If the vector A does not change in the moving system, its rate of change in the fixed system is due only to the rotation, so that dA/dt = SxA; see §9, where it has been pointed out that formulae such as (9.1) and (9.2) are valid for any vector. In the general case, the right-hand side includes also the rate of change of the vector A with respect to the moving system. Denoting this rate of change by d’A/dt, we obtain dAdd (36.1) §36 Euler’s equations 115 Using this general formula, we can immediately write equations (34.1) and (34.3) in the form = K. (36.2) Since the differentiation with respect to time is here performed in the moving system of co-ordinates, we can take the components of equations (36.2) along the axes of that system, putting (d’P/dt)1 = dP1/dt, …, (d’M/dt)1 = dM1/dt, …, where the suffixes 1, 2, 3 denote the components along the axes x1, x2, X3. In the first equation we replace P by V, obtaining (36.3) = If the axes X1, X2, X3 are the principal axes of inertia, we can put M1 = I, etc., in the second equation (36.2), obtaining = I2 = K2, } (36.4) I3 = K3. These are Euler’s equations. In free rotation, K = 0, so that Euler’s equations become = 0, } (36.5) = 0. As an example, let us apply these equations to the free rotation of a sym- metrical top, which has already been discussed. Putting I1 = I2, we find from the third equation SQ3 = 0, i.e. S3 = constant. We then write the first two equations as O = -wS2, Q2 = wS1, where = (36.6) is a constant. Multiplying the second equation by i and adding, we have = so that S1+iD2 = A exp(iwt), where A is a constant, which may be made real by a suitable choice of the origin of time. Thus S1 = A cos wt Q2 = A sin wt. (36.7) 116 Motion of a Rigid Body