2.8 KiB
2.8 KiB
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| 36-eulers-equations |
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