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\title{
A numerical scheme to integrate the rotational motion
of a quasi-rigid body and its application to the Earth rotation
}
\author{ TOSHIO FUKUSHIMA \\ 
National Astronomical Observatory \\
2-21-1, Ohsawa, Mitaka, Tokyo 181, JAPAN \\
(Internet) toshio@spacetime.mtk.nao.ac.jp
}
\date{}
\begin{document}
\maketitle
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\noindent
A new numerical scheme to integrate the orientation of
a quasi-rigid body is presented.
The basic variables are
\[
(L, L_{\rm X}, L_{\rm Y}, L_{\rm A}, L_{\rm B}, f)
\]
the magnitude and the $X$-, $Y$-, $A$-, and $B$-components
of the spin angular momentum vector ${\bf L}$,
and the longitude of $A$-axis measured from $X$-axis
along the great circle  perpendicular to ${\bf L}$.
The orientation matrix
is generated by the five successive rotations
in the sense of 1-2-3-2-1 as
\[
({\bf e}_{\rm A}, {\bf e}_{\rm B}, {\bf e}_{\rm C})
= {\cal R}_1 (\sigma)
{\cal R}_2 (- \varphi)
{\cal R}_3 (- f)
{\cal R}_2 (\beta)
{\cal R}_1 (- \xi)
\]
where
\[
\sigma = \tan^{-1} \left(\frac{L_{\rm Y}}
   {\sqrt{L^2-(L_{\rm X}^2+L_{\rm Y}^2)}}\right),
\hspace{3em}
\varphi = \sin^{-1} \frac{L_{\rm X}}{L},
\]
\[
\beta= \sin^{-1} \frac{L_{\rm A}}{L},
\hspace{3em}
\xi= \tan^{-1} \left(\frac{L_{\rm B}}
   {\sqrt{L^2 - (L_{\rm A}^2 + L_{\rm B}^2)}}\right).
\]
Not all these basic variables but the corrections
to their nominal linear motions for some components instead
such as
\[
({\sl \Delta} L, {\sl \Delta} L_{\rm X}, {\sl \Delta} L_{\rm Y},
L_{\rm A}, L_{\rm B}, {\sl \Delta} f)
\]
are actually integrated in the inertial coordinate system
whose $Z$-axis is chosen to be close to ${\bf L}$ at the
initial epoch of integration.
These small variables directly correspond to the variation of LOD,
the nutations in obliquity and in declination, the polar motion,
and the variation of UT1 in terms of the Earth rotation.
Numerical simulations showed that the new scheme integrates
the Earth orientation matrix 6-7 digits more precisely
than the ordinary Eulerian approach does
while the required computational time does not change significantly.
\end{document}
