This result shows that the component of the angular velocity perpendicular to the axis of the top rotates with an angular velocity w, remaining of constant magnitude A = Since the component S3 along the axis of the top is also constant, we conclude that the vector S rotates uniformly with angular velocity w about the axis of the top, remaining unchanged in magni- tude. On account of the relations M1 = , M2 = I2O2, M3 = I3O3 be- tween the components of S and M, the angular momentum vector M evidently executes a similar motion with respect to the axis of the top. This description is naturally only a different view of the motion already discussed in §33 and §35, where it was referred to the fixed system of co- ordinates. In particular, the angular velocity of the vector M (the Z-axis in Fig. 48, $35) about the x3-axis is, in terms of Eulerian angles, the same as the angular velocity - 4. Using equations (35.4), we have cos or - is = I23(I3-I1)/I1, in agreement with (36.6). §37. The asymmetrical top We shall now apply Euler’s equations to the still more complex problem of the free rotation of an asymmetrical top, for which all three moments of inertia are different. We assume for definiteness that I3 > I2 I. (37.1) Two integrals of Euler’s equations are known already from the laws of conservation of energy and angular momentum: = 2E, (37.2) = M2, where the energy E and the magnitude M of the angular momentum are given constants. These two equations, written in terms of the components of the vector M, are (37.3) M2. (37.4) From these equations we can already draw some conclusions concerning the nature of the motion. To do so, we notice that equations (37.3) and (37.4), regarded as involving co-ordinates M1, M2, M3, are respectively the equation of an ellipsoid with semiaxes (2EI1), (2EI2), (2EI3) and that of a sphere of radius M. When the vector M moves relative to the axes of inertia of the top, its terminus moves along the line of intersection of these two surfaces. Fig. 51 shows a number of such lines of intersection of an ellipsoid with §37 The asymmetrical top 117 spheres of various radii. The existence of an intersection is ensured by the obviously valid inequalities 2EI1 < M2 < 2EI3, (37.5) which signify that the radius of the sphere (37.4) lies between the least and greatest semiaxes of the ellipsoid (37.3). x1 X2 FIG. 51 Let us examine the way in which these “paths”t of the terminus of the vector M change as M varies (for a given value of E). When M2 is only slightly greater than 2EI1, the sphere intersects the ellipsoid in two small closed curves round the x1-axis near the corresponding poles of the ellipsoid; as M2 2EI1, these curves shrink to points at the poles. When M2 increases, the curves become larger, and for M2 = 2EI2 they become two plane curves (ellipses) which intersect at the poles of the ellipsoid on the x2-axis. When M2 increases further, two separate closed paths again appear, but now round the poles on the x3-axis; as M2 2EI3 they shrink to points at these poles. First of all, we may note that, since the paths are closed, the motion of the vector M relative to the top must be periodic; during one period the vector M describes some conical surface and returns to its original position. Next, an essential difference in the nature of the paths near the various poles of the ellipsoid should be noted. Near the x1 and X3 axes, the paths lie entirely in the neighbourhood of the corresponding poles, but the paths which pass near the poles on the x2-axis go elsewhere to great distances from those poles. This difference corresponds to a difference in the stability of the rota- tion of the top about its three axes of inertia. Rotation about the x1 and X3 axes (corresponding to the least and greatest of the three moments of inertia) t The corresponding curves described by the terminus of the vector Ca are called polhodes. 118 Motion of a Rigid Body §37 is stable, in the sense that, if the top is made to deviate slightly from such a state, the resulting motion is close to the original one. A rotation about the x2-axis, however, is unstable: a small deviation is sufficient to give rise to a motion which takes the top to positions far from its original one. To determine the time dependence of the components of S (or of the com- ponents of M, which are proportional to those of (2) we use Euler’s equations (36.5). We express S1 and S3 in terms of S2 by means of equations (37.2) and (37.3): S21 = (37.6) Q32 = and substitute in the second equation (36.5), obtaining dSQ2/dt (I3-I1)21-23/I2 = V{[(2EI3-M2-I2(I3-I2)22] (37.7) Integration of this equation gives the function t(S22) as an elliptic integral. In reducing it to a standard form we shall suppose for definiteness that M2 > 2EI2; if this inequality is reversed, the suffixes 1 and 3 are interchanged in the following formulae. Using instead of t and S2 the new variables (37.8) S = S2V[I2(I3-I2)/(2EI3-M2)], and defining a positive parameter k2 < 1 by (37.9) we obtain ds the origin of time being taken at an instant when S2 = 0. When this integral is inverted we have a Jacobian elliptic function S = sn T, and this gives O2 as a function of time; S-1(t) and (33(t) are algebraic functions of 22(t) given by (37.6). Using the definitions cn T = V(1-sn2r), dn T = we find Superscript(2) = [(2EI3-M2/I1(I3-I1)] CNT, O2 = (37.10) O3 = dn T. These are periodic functions, and their period in the variable T is 4K, where K is a complete elliptic integral of the first kind: = (37.11) §37 The asymmetrical top 119 The period in t is therefore T = (37.12) After a time T the vector S returns to its original position relative to the axes of the top. The top itself, however, does not return to its original position relative to the fixed system of co-ordinates; see below. For I = I2, of course, formulae (37.10) reduce to those obtained in §36 for a symmetrical top: as I I2, the parameter k2 0, and the elliptic functions degenerate to circular functions: sn -> sin T, cn T cos T, dn T -> 1, and we return to formulae (36.7). When M2 = 2EI3 we have Superscript(1) = S2 = 0, S3 = constant, i.e. the vector S is always parallel to the x3-axis. This case corresponds to uniform rotation of the top about the x3-axis. Similarly, for M2 = 2EI1 (when T III 0) we have uniform rotation about the x1-axis. Let us now determine the absolute motion of the top in space (i.e. its motion relative to the fixed system of co-ordinates X, Y, Z). To do so, we use the Eulerian angles 2/5, o, 0, between the axes X1, X2, X3 of the top and the axes X, Y, Z, taking the fixed Z-axis in the direction of the constant vector M. Since the polar angle and azimuth of the Z-axis with respect to the axes x1, X2, X3 are respectively 0 and 1/77 - is (see the footnote to $35), we obtain on taking the components of M along the axes X1, X2, X3 M sin 0 sin y = M1 = , M sin A cos is = M2 = I2O2, (37.13) M cos 0 = M3 = I3S23. Hence cos 0 = I3S3/M, tan / = (37.14) and from formulae (37.10) COS 0 = dn T, (37.15) tan 4 = cn r/snt, which give the angles 0 and is as functions of time; like the components of the vector S, they are periodic functions, with period (37.12). The angle does not appear in formulae (37.13), and to calculate it we must return to formulae (35.1), which express the components of S in terms of the time derivatives of the Eulerian angles. Eliminating O from the equa- tions S1 = sin 0 sin 4 + O cos 2/5, S2 = sin 0 cos 4-0 - sin 2/5, we obtain & = (Superscript(2) sin 4+S2 cos 4)/sin 0, and then, using formulae (37.13), do/dt = (37.16) The function (t) is obtained by integration, but the integrand involves elliptic functions in a complicated way. By means of some fairly complex 120 Motion of a Rigid Body §37 transformations, the integral can be expressed in terms of theta functions; we shall not give the calculations, but only the final result. The function (t) can be represented (apart from an arbitrary additive constant) as a sum of two terms: $(t) = (11(t)++2(t), (37.17) one of which is given by (37.18) where D01 is a theta function and a a real constant such that sn(2ixK) = iv[I3(M2-2I1)/I1(2EI3-M2] (37.19) K and Tare given by (37.11) and (37.12). The function on the right-hand side of (37.18) is periodic, with period 1T, so that 01(t) varies by 2n during a time T. The second term in (37.17) is given by (37.20) This function increases by 2nr during a time T’. Thus the motion in is a combination of two periodic motions, one of the periods (T) being the same as the period of variation of the angles 4 and 0, while the other (T’) is incom- mensurable with T. This incommensurability has the result that the top does not at any time return exactly to its original position. PROBLEMS PROBLEM 1. Determine the free rotation of a top about an axis near the x3-axis or the x1-axis. SOLUTION. Let the x3-axis be near the direction of M. Then the components M1 and M2 are small quantities, and the component M3 = M (apart from quantities of the second and higher orders of smallness). To the same accuracy the first two Euler’s equations (36.5) can be written dM1/dt = DoM2(1-I3/I2), dM2/dt = QOM1(I3/I1-1), where So = M/I3. As usual we seek solutions for M1 and M2 proportional to exp(iwt), obtaining for the frequency w (1) The values of M1 and M2 are cos wt, sin wt, (2) where a is an arbitrary small constant. These formulae give the motion of the vector M relative to the top. In Fig. 51, the terminus of the vector M describes, with frequency w, a small ellipse about the pole on the x3-axis. To determine the absolute motion of the top in space, we calculate its Eulerian angles. In the present case the angle 0 between the x3-axis and the Z-axis (direction of M) is small, t These are given by E. T. WHITTAKER, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, 4th ed., Chapter VI, Dover, New York 1944. §37 The asymmetrical top 121 and by formulae (37.14) tan of = M1/M2, cos 0) 2(1 (M3/M) 22 substituting (2), we obtain tan 4 = V[I(I3-I2)/I2(I3-I1)] cot wt, (3) To find , we note that, by the third formula (35.1), we have, for 0 1, Hence = lot (4) omitting an arbitrary constant of integration. A clearer idea of the nature of the motion of the top is obtained if we consider the change in direction of the three axes of inertia. Let n1, n2, n3 be unit vectors along these axes. The vectors n1 and n2 rotate uniformly in the XY-plane with frequency So, and at the same time execute small transverse oscillations with frequency w. These oscillations are given by the Z-components of the vectors: 22 M1/M = av(I3/I2-1) cos wt, N2Z 22 M2/M = av(I3/I1-1) sin wt. For the vector n3 we have, to the same accuracy, N3x 22 0 sin , N3y 22 -0 cos , n3z 1. (The polar angle and azimuth of n3 with respect to the axes X, Y, Z are 0 and -; see the footnote to 35.) We also write, using formulae (37.13), naz=0sin(Qot-4) = Asin Sot cos 4-0 cos lot sin 4 = (M 2/M) sin Dot-(M1/M) cos Sot sin Sot sin N/1-1) cos Not cos wt cos(so Similarly From this we see that the motion of n3 is a superposition of two rotations about the Z-axis with frequencies So + w. PROBLEM 2. Determine the free rotation of a top for which M2 = 2EI2. SOLUTION. This case corresponds to the movement of the terminus of M along a curve through the pole on the x2-axis (Fig. 51). Equation (37.7) becomes ds/dr = 1-s2, = S = I2/20, where So = M/I2 = 2E|M. Integration of this equation and the use of formulae (37.6) gives sech T, } (1) sech T. To describe the absolute motion of the top, we use Eulerian angles, defining 0 as the angle between the Z-axis (direction of M) and the x2-axis (not the x3-axis as previously). In formulae (37.14) and (37.16), which relate the components of the vector CA to the Eulerian angles, we 5 122 Motion of a Rigid Body