§35
Eulerian angles
111
Let us now express the components of the angular velocity vector S along
the moving axes X1, X2, X3 in terms of the Eulerian angles and their derivatives.
To do this, we must find the components along those axes of the angular
velocities 6, b, 4. The angular velocity è is along the line of nodes ON, and
its components are 1 = O cos 4/5, = - O sin 4/5, = 0. The angular velo-
city is along the Z-axis; its component along the x3-axis is 03 = cos 0, and
in the x1x2-plane sin A. Resolving the latter along the X1 and X2 axes, we
have 01 = sin 0 sin 4/s, O2 = sin 0 cos 4. Finally, the angular velocity is
is along the x3-axis.
Collecting the components along each axis, we have
S21 = 0 COS 4,
Q2 = sin 0 cosy-osiny,
(35.1)
S23 = o cos0+4. =
If the axes X1, X2, X3 are taken to be the principal axes of inertia of the body,
the rotational kinetic energy in terms of the Eulerian angles is obtained by
substituting (35.1) in (32.8).
For a symmetrical top (I1 = I2 # I3), a simple reduction gives
Trot =
(35.2)
This expression can also be more simply obtained by using the fact that the
choice of directions of the principal axes X1, X2 is arbitrary for a symmetrical
top. If the X1 axis is taken along the line of nodes ON, i.e. 4 = 0, the compo-
nents of the angular velocity are simply
O2 = o sin A,
(35.3)
As a simple example of the use of the Eulerian angles, we shall use them
to determine the free motion of a symmetrical top, already found in $33.
We take the Z-axis of the fixed system of co-ordinates in the direction of the
constant angular momentum M of the top. The x3-axis of the moving system
is along the axis of the top; let the x1-axis coincide with the line of nodes at
the instant considered. Then the components of the vector M are, by
formulae (35.3), M1 = I1 = I, M2 = IS2 = sin 0, M3 = I3Q3
= I3( cos 0+4). Since the x1-axis is perpendicular to the Z-axis, we have
M1 = 0, M2 = M sin 0, M3 = M cos 0. Comparison gives
0=0,
I = M,
=
(35.4)
The first of these equations gives 0 = constant, i.e. the angle between the
axis of the top and the direction of M is constant. The second equation gives
the angular velocity of precession = M/I1, in agreement with (33.5).
Finally, the third equation gives the angular velocity with which the top
rotates about its own axis: S3 = (M/I3) cos 0.
112
Motion of a Rigid Body
§35
PROBLEMS
PROBLEM 1. Reduce to quadratures the problem of the motion of a heavy symmetrical
top whose lowest point is fixed (Fig. 48).
SOLUTION. We take the common origin of the moving and fixed systems of co-ordinates
at the fixed point O of the top, and the Z-axis vertical. The Lagrangian of the top in a gravita-
tional field is L = (02 +02 sin ²0 + 1/3(1- cos 0)2-ugl - cos 0, where u is the mass
of the top and l the distance from its fixed point to the centre of mass.
Z
X3
x2
a
ug
Y
x1
N
FIG. 48
The co-ordinates 4 and are cyclic. Hence we have two integrals of the motion:
P4 = = cos 0) = constant = M3
(1)
= = cos 0 = constant III M2,
(2)
where I'1 = I1+ul2; the quantities P4 and Po are the components of the rotational angular
momentum about O along the X3 and Z axes respectively. The energy
E = cos 0
(3)
is also conserved.
From equations (1) and (2) we find
=
0)/I'1
sin 20,
(4)
(5)
Eliminating b and of from the energy (3) by means of equations (4) and (5), we obtain
E' =
where
E'
=
(6)
§35
Eulerian angles
113
Thus we have
t=
(7)
this is an elliptic integral. The angles 4 and are then expressed in terms of 0 by means of
integrals obtained from equations (4) and (5).
The range of variation of 0 during the motion is determined by the condition E' Ueff(0).
The function Uett(8) tends to infinity (if M3 # M2) when 0 tends to 0 or II, and has a minimum
between these values. Hence the equation E' = Ueff(0) has two roots, which determine the
limiting values 01 and O2 of the inclination of the axis of the top to the vertical.
When 0 varies from 01 to O2, the derivative o changes sign if and only if the difference
M-M3 cos 0 changes sign in that range of 0. If it does not change sign, the axis of the top
precesses monotonically about the vertical, at the same time oscillating up and down. The
latter oscillation is called nutation; see Fig. 49a, where the curve shows the track of the axis
on the surface of a sphere whose centre is at the fixed point of the top. If does change sign,
the direction of precession is opposite on the two limiting circles, and so the axis of the top
describes loops as it moves round the vertical (Fig. 49b). Finally, if one of 01, O2 is a zero of
M2-M3 cos 0, of and è vanish together on the corresponding limiting circle, and the path
of the axis is of the kind shown in Fig. 49c.
O2
O2
O2
(a)
(b)
(c)
FIG. 49
PROBLEM 2. Find the condition for the rotation of a top about a vertical axis to be stable.
SOLUTION. For 0 = 0, the X3 and Z axes coincide, so that M3 = Mz, E' = 0. Rotation
about this axis is stable if 0 = 0 is a minimum of the function Ueff(9). For small 0 we have
Ueff 22 whence the condition for stability is M32 > 41'1ugl or S232
> 41'1ugl/I32.
PROBLEM 3. Determine the motion of a top when the kinetic energy of its rotation about
its axis is large compared with its energy in the gravitational field (called a "fast" top).
SOLUTION. In a first approximation, neglecting gravity, there is a free precession of the
axis of the top about the direction of the angular momentum M, corresponding in this case
to the nutation of the top; according to (33.5), the angular velocity of this precession is
Sunu = M/I' 1.
(1)
In the next approximation, there is a slow precession of the angular momentum M about
the vertical (Fig. 50). To determine the rate of this precession, we average the exact equation
of motion (34.3) dM/dt = K over the nutation period. The moment of the force of gravity
on the top is K=uln3xg, where n3 is a unit vector along the axis of the top. It is evident
from symmetry that the result of averaging K over the "nutation cone" is to replace n3 by
its component (M/M) cos a in the direction of M, where a is the angle between M and the
axis of the top. Thus we have dM/dt = -(ul/M)gxM cos a. This shows that the vector M
114
Motion of a Rigid Body
§36
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
§37
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.