2. December 01, 2009
10.7 Euler Equations
We now know that all objects have three principal axes, and that we can
always write the inertia tensor as a diagonal matrix relative to these axes,
so that the angular momentum vector can be written L = (l1w1, l2w2, l3w3).
In the accelerating (rotating) frame of the body (we will call it the body
frame), the body is stationary with respect to the rotation. The inertial
(non-rotating) frame is traditionally called the space frame.
We can use coordinates x, y, z as the coordinates in the space frame, and e1,
e2, e3 as unit vectors in the directions of the principle axes in the body
frame.
Recall that the time derivative of the angular momentum is the torque:
We can transform to the body frame by adding the usual w × L:
This leads us to Euler’s equation:
.
space
Γ
L
dt
d
.
body
space
L
ω
L
L
dt
d
dt
d
.
Γ
L
ω
L
3. December 01, 2009
Euler Equations-2
Since L = (l1w1, l2w2, l3w3), we have .
Therefore, the Euler vector equation can be written as three
equations:
These equations give the evolution of w in the frame of the body, but in
general they are difficult to use because they require knowledge of the
components G1, G2 and G3 of the applied torque, which in the rotating frame
are complicated.
They are useful, however, in cases where the torque is zero, and that is the
case we will discuss further.
Meanwhile, for the example of the spinning top we have the case that the
torque is always perpendicular to e3, so G3 is zero. Also, because of the
symmetry of the top, l1 = l2. The third equation above then says
This means the component of w along the symmetry axis is constant.
.
)
(
)
(
)
(
3
2
1
2
1
3
3
2
3
1
1
3
2
2
1
3
2
3
2
1
1
G
G
G
w
w
l
l
w
l
w
w
l
l
w
l
w
w
l
l
w
l
).
,
,
( 3
3
2
2
1
1 w
l
w
l
w
l
L
Γ
L
ω
L
.
0
3
3
w
l
4. December 01, 2009
10.8 Euler Equations with Zero Torque
In the case of zero torque, Euler’s Equations become:
We will first consider these equations when the three principal moments are
all different, and then for the case, as for the spinning top, that l1 = l2.
One thing we can see immediately is that if the rotation axis starts out
pointing along one of the principal axes (e.g. w = wi), it will stay in that
direction. That is, two of the equations will be zero, and the third (ith)
equation says w = constant.
Conversely, if the rotation axis is NOT along one of the principle axes, its
direction will not be constant. To see this, note that at least two of the
components of w must be non-zero, which means at least one component of
must be non-zero.
So, rotation about a principal axis is the only way to have a constant rotation
axis. An interesting question is whether such a rotation is stable—that is, if
we slightly perturb the rotation will it grow with time?
1 1 2 3 2 3
2 2 3 1 1 3
3 3 1 2 1 2
( )
( ) .
( )
l w l l w w
l w l l w w
l w l l w w
0
i
w
ω
5. December 01, 2009
Stability Against Perturbation
To answer this, let’s assume we initially have rotation about the 3 axis, so
that w1 = w2 = 0. If we give the body a small tap, so that both of these
become non-zero, but small, then the third Euler equation says will be
“doubly small”. If we take it to be constant, then we have only the first two
equations
where the quantities in square brackets are (approximately) constant.
As usual, we can take the time derivative of the first and substitute into the
second to get
You can recognize this immediately as simple harmonic motion in w1 with the
oscillation frequency
But note that the radical could be negative, which means w is imaginary.
1 1 2 3 3 2
2 2 3 1 3 1
( )
,
( )
l w l l w w
l w l l w w
3
w
2
3 2 3 1
1 3 1
1 2
( )( )
.
l l l l
w w w
l l
3 2 3 1
3
1 2
( )( )
.
l l l l
w w
l l
Stable if l3 > l1 and l2
or if l3 < l1 and l2
6. December 01, 2009
Stability Against Perturbation-2
We therefore have the interesting situation that if you spin an object about
either its longest or shortest axes, it will be stable, but around its intermediate
axis, it is unstable.
Stability of a Top
Now let’s consider the case, as for the spinning top, that l1 = l2. In this case,
we have
Then we can write the other two Euler equations in terms of a constant
frequency Wb:
stable
stable
unstable
constant.
or
0 3
3
w
w
w
w
l
w
l
l
w
w
w
l
w
l
l
w
b
1
b
1
1
3
3
1
2
2
b
2
1
3
3
1
1
or
)
(
)
(
W
W
W
i
= w1 + iw2
7. December 01, 2009
Stability of a Top
This has solution:
Let’s take initial conditions w1 = wo and w2 = 0 at t = 0. Then o = wo, and
when we put in the real and imaginary parts of , the complete solution is:
with wo and w3 both constant.
The relationship between w1 and w2 is that of a rotating (precessing) vector
component, precessing with frequency Wb. Therefore, as seen from the body
frame (the one rotating with angular frequency w3 in the direction e3), the
vector w moves steadily around a cone, called the body cone., centered on
the direction e3.
Meanwhile, the angular momentum L is give by
In the space frame, w precesses about L. at a
different frequency, Wb = L/l1.
This is called free precession, and does not
require a torque.
)
,
sin
,
cos
( 3
b
b w
w
w t
t o
o W
W
ω
.
bt
i
oe W
).
,
sin
,
cos
(
)
,
,
(
3
3
b
1
b
1
3
3
2
1
1
1
w
l
w
l
w
l
w
l
w
l
w
l
t
t o
o W
W
L
L
e3 (fixed)
w
L (fixed)
e3
w
body frame space frame
Wb
Ws
= w1 + iw2
