Dynamics of an isolated, viscoelastic, self-gravitating body

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This paper is devoted to an alternative model for a rotating, isolated, self-gravitating, viscoelastic body. The initial approach is quite similar to the classical one, present in the works of Dirichlet, Riemann, Chandrasekhar, among others. Our main contribution is to present a simplified model for the motion of an almost spherical body. The Lagrangian function \({\fancyscript{L}}\) and the dissipation function \({\fancyscript{D}}\) of the simplified model are: $$\begin{aligned} {\fancyscript{L}}=\frac{\omega \cdot \mathrm{I}\omega }{2}+ \frac{1}{36\, \mathrm{I}_\circ }(\Vert \dot{Q}\Vert ^2-\gamma \Vert Q\Vert ^2) \end{aligned}$$ and $$\begin{aligned} {\fancyscript{D}}=\frac{\nu }{36\, \mathrm{I}_\circ }\Vert \dot{Q}\Vert ^2 \end{aligned}$$ where \(\omega \) is the angular velocity vector, \(Q\) is the quadrupole moment tensor, \(\mathrm{I}=\mathrm{I}_\circ \mathrm{\!\ \mathbb {I}d\!\ } -Q/3\) is the usual moment of inertia tensor with \(\mathrm{I}_\circ \) equal to the moment of inertia of the spherical body at rest, \(\gamma \) is an elastic constant, and \(\nu \) is a damping coefficient. The angular momentum \(\mathrm{I}\omega \) transformed to an inertial reference frame is conserved. The constants \(\gamma \) and \(\nu \) must be determined experimentally. We believe this to be the simplest model one can get without loosing the symmetries and the conserved quantities of the original problem. This model can be used as a building block for the study of many-body planetary systems.