Transformation of matrix equations for perturbed Earth’s rotation into excitation equations

A convenient method for studying the influence of small perturbations (e.g., the effect of oceans, atmosphere, and hydrological factors) on the Earth’s rotation consists in representing the Earth’s rotation equation in the form of excitation equations introduced by Walter Munk and Gordon MacDonald. Traditionally, small perturbations are taken into account in two steps: first, an exact solution is found for equations describing the unperturbed rotation of the adopted Earth model; then, the effect of small perturbations is taken into account in the form of corrections to this exact solution. In most modern and probably future theories of the Earth’s rotation, the equations for unperturbed rotation can be written in the form of linear matrix differential equations. The present study aims to take into account the effect of small perturbations to the Earth’s rotation in new theories of the Earth’s rotation. An algorithm was developed and presented for transforming matrix linear differential equations for perturbed rotation into excitation equations. As an example, the proposed transformation algorithm was applied to the relatively simple equations of the Sasao-Okubo-Saito theory derived from the original Molodensky Earth model. The equations for finding corrections to the original solution of the Euler-Liouville equations are reduced to the form of excitation equations. The developed algorithm is proposed to be used for a transition from the matrix equations for unperturbed Earth’s rotation to the excitation equations in new theories of Earth’s rotation.