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Astronomical Applications Department, U.S. Naval Observatory - Precession Maple (Page 12)

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Astronomical Applications Department, U.S. Naval Observatory - Precession Maple
Hence,
=
+




t
(
)
cos




t
const
This is the projection of the angular velocity onto the symmetry axis. Recall the components
of
in the body frame:
=
(
)
mat
,
,
x
y
z
(
)
convert
,
Obody matrix
=


x
y
z


+




t
( )
sin
(
)
sin




t
( )
cos
-




t
( )
cos
(
)
sin




t
( )
sin
+




t
(
)
cos




t
We see that the angular velocity about the symmetry axis,
z
, is a constant of the motion. We
also have conservation of energy,
=
E
+
+
I
x
x
2
I
y
y
2
I
z
z
2
2
:=
KE
%
Substituting the components of
in terms of the Euler angles, we have




collect
,
,




subs
,




seq
,
=
( )
xyz
k
t
Obody
k
=
k
..
1
3
KE
[
]
,
diff
(
)
sin
factor
:=
KE
%
E
1
2
I
z




t
2
I
z




t
(
)
cos




t
+
=




+




+
1
2
I
x
( )
sin
2
1
2
I
y
( )
cos
2
(
)
sin
2
1
2
I
z
(
)
cos
2




t
2
+
( )
cos
( )
sin
(
)
- +
I
y
I
x
(
)
sin




t




t




+
1
2
I
x
( )
cos
2
1
2
I
y
( )
sin
2




t
2
+
+
:=
subslist




,
,
=
t
=
t
=
t
(
)
subs
,
,
,
subslist
=
I
x
I
y
=
I
y
I
xy
KE
E
1
2
I
z
2
I
z
(
)
cos
+
=




+




+
1
2
I
xy
( )
sin
2
1
2
I
xy
( )
cos
2
(
)
sin
2
1
2
I
z
(
)
cos
2
2
+




+
1
2
I
xy
( )
sin
2
1
2
I
xy
( )
cos
2
2
+
Page 12

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