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\author{Juan Getino\thanks{Grupo de Mec nica Celeste, Departamento de Matem\'atica Aplicada Fundamental,
Facultad de Ciencias, Universidad de Valladolid, E-47005 Valladolid, Spain.}
\and
Luis Flor\'{\i}a\thanks{Grupo de Mec nica Celeste, Departamento de Matem\'atica Aplicada a la
Ingenier\'{\i}a, E.T.S. de Ingenieros In\-dus\-tria\-les, Universidad de Valladolid,
E-47011 Valladolid, Spain.}}
\date{}
\title{Canonical treatment of dissipative forces between Earth core and mantle}
\maketitle
\begin{abstract}
Getino and Ferr\'andiz (1995) applied a Hamiltonian approach to the
study of the effect of the deformations of an elastic mantle on the Earth
rotation. Later, Getino (1995a, 1995b) considered, in a similar way, the
problem of the liquid core and the influence of the inertial interaction
term. As a particular merit, these works point out a possibility of
extending the canonical theory for the rigid Earth developed by Kinoshita
(1977) to the case of
a deformable Earth. This can be accomplished by following the same procedure,
and then, taking advantage
of the simplicity of such a powerful formulation.

In this paper we perform a new step in our investigation, dealing with the
interaction due to the frictional forces between mantle and core. First, the
dissipative forces are formulated in a canonical form starting from
the procedure of
Kubo (1979), but using the more suitable set of canonical variables
worked out by Getino (1995a, 1995b). Next, we carry out a canonical
integration of the Hamiltonian following the guidelines of some canonical
perturbation methods for non-conservative systems such as those of
Choi and Tapley (1973) and Tveter (1994). A numerical estimation shows
that the effect of
the friction on the amplitude of the nutation is small. However, due to these
forces, phase angles appear. This effect is not obtained by the well-known
theory of Wahr (1981).

We emphasize that, at this stage of the investigation, the main aim of this
paper is not the obtaining of values for the amplitude and the phase
of the nutation
according to the observational data, but to extend the canonical formalism
to a more realistic Earth model, in order to get in the future a
complete Hamiltonian
theory explaining the most important effects concerning the Earth rotation.
\end{abstract}
\noindent
{\bf Key words}: Geopotential, Solid tides, Elastic Earth.
\begin{thebibliography}{9}
\bibitem{} Choi, J.S. and Tapley, B.D.: 1973, {\it Celes. Mech.}, {\bf 7},
77--90.

\bibitem{} Getino, J., Ferr\'andiz, J.M.: 1995, {\it Celes. Mech.}, {\bf 61}%
, 117--180.

\bibitem{} Getino, J.: 1995a, ``An interpretation of the core--mantle
interaction problem'', {\it Geophys. J. Int.} (in press).

\bibitem{} Getino, J.: 1995b, ``Forced nutations of a rigid mantle--liquid
core earth model in canonical formulation'', Geophys. J. Int. (submitted).

\bibitem{} Kinoshita, H.: 1977, {\it Celes. Mech.}, {\bf 15}, 277--326.

\bibitem{} Kubo, Y.: 1979, {\it Celes. Mech.}, {\bf 19}, 215--241.

\bibitem{} Tveter, F.K.: 1994, {\it Celes. Mech.}, {\bf 60}, 409--419.
\end{thebibliography}

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