Hamiltonian mechanics is applied to the problem of the rotation of the elastic Earth. We first show the process for the formulation of the Hamiltonian for rotation of a deformable body and the derivation of the equations of motion from it. Then, based on a simple model of deformation, the solution is given for the period of Euler motion, UT1 and the nutation of the elastic Earth. In particular it is shown that the elasticity of the Earth acts on the nutation so as to decrease the Oppolzer terms of the nutation of the rigid Earth by about 30 per cent. The solution is in good agreement with results which have been obtained by other, different approaches.

We calculate the so-called convective term, which shows up in the expression for the angular velocity of the elastic Earth, within the Andoyer formalism. The term emerges due to the fact that the elasticity-caused perturbation depends not only on the instantaneous orientation of the Earth but also on its instantaneous angular velocity. We demonstrate that this term makes a considerable contribution into the overall angular velocity. At the same time the convective term turns out to be automatically included into the correction to the nutation series due to the elasticity, if the series is defined by the perturbation of the figure axis (and not of the rotational axis) in accordance with the current IAU resolution. Hence it is not necessary to take the effect of the convective term into consideration in the perturbation of the elastic Earth as far as the nutation is related to the motion of the figure axis.

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Under perturbations from outer bodies, the Earth experiences changes of its angular momentum axis, figure axis and rotational axis. In the theory of the rigid Earth, in addition to the precession and nutation of the angular momentum axis given by the Poisson terms, both the figure axis and the rotational axis suffer forced deviation from the angular momentum axis. This deviation is expressed by the so-called Oppolzer terms describing separation of the averaged figure axis, called CIP (Celestial Intermediate Pole) or CEP (Celestial Ephemeris Pole), and the mathematically defined rotational axis, from the angular momentum axis. The CIP is the rotational axis in a frame subject to both precession and nutation, while the mathematical rotational axis is that in the inertial (non-rotating) frame. We investigate, kinematically, the origin of the separation between these two axes-both for the rigid Earth and an elastic Earth. In the case of an elastic Earth perturbed by the same outer bodies, there appear further deviations of the figure and rotational axes from the angular momentum axis. These deviations, though similar to the Oppolzer terms in the rigid Earth, are produced by quite a different physical mechanism. Analysing this mechanism, we derive an expression for the Oppolzer-like terms in an elastic Earth. From this expression we demonstrate that, under a certain approximation (in neglect of the motion of the perturbing outer bodies), the sum of the direct and convective perturbations of the spin axis coincides with the direct perturbation of the figure axis. This equality, which is approximate, gets violated when the motion of the outer bodies is taken into account.