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
