[1] The analytical formulation of the theories of nutation and wobble reveals the combinations of basic Earth parameters that govern the nutation-wobble response of the Earth to gravitational (tidal) forcing by heavenly bodies and makes it possible to estimate several of them through a least squares fit of the theoretical expressions to the high-precision data now available. This paper presents the essentials of the theoretical framework, the procedure that we used for least squares estimation of basic Earth parameters through a fit of theory to nutation-precession data derived from an up-to-date very long baseline interferometry data set, the results of the estimation and their geophysical interpretation, and the nutation series constructed using the estimated values of the parameters. The theoretical formulation used here differs from earlier ones in the incorporation of anelasticity and ocean tide effects into the basic structure of the dynamical equations of the theory and in the inclusion of electromagnetic couplings of the mantle and the solid inner core to the fluid outer core, though this generalization comes at the cost of making some of the system parameters complex and frequency dependent; it is also more complete, as it takes account of nonlinear terms in these equations, including effects of the time-dependent deformations produced by zonal and sectorial tides, which had been traditionally neglected in nonrigid Earth theories. Among the geophysical results obtained from our fit are estimates for the dynamic ellipticity e of the Earth (e = 0.0032845479 with an uncertainty of 12 in the last digit), for the dynamical ellipticity e f of the fluid core (3.8% higher than its hydrostatic equilibrium value, rather than $5% as hitherto), and for the two complex electromagnetic coupling constants. Our best estimates for the RMS radial magnetic fields at the core mantle boundary and at the inner core boundary, based on the estimates for these coupling constants, are~6.9 and 72 gauss, respectively, when the magnetic field configurations are restricted to certain simple classes. The field strength needed at the inner core boundary could be lower if the density of the core fluid at this boundary or the ellipticity of the solid inner core were lower than that for the Preliminary Reference Earth Model. Our estimate for the resonance frequency of the prograde free core nutation mode, with an uncertainty of $10%, constitutes the first firm detection of the resonance associated with this mode; the period found is $1025 days, double that with electromagnetic couplings ignored. (Throughout this work, ''days,'' referring to periods, stands for ''mean solar days.'') A new nutation series (MHB2000) is constructed by direct solution of the linearized dynamical equations (with our best fit values adopted for all the estimated Earth parameters) for each forcing frequency, and adding on the contributions from the nonlinear terms and other effects not included in the linearized equations. This series gives a considerably better...

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Gravitational and pressure couplings between the solid inner core and the rest of the Earth give rise to torques through which the inner core influences the nutational motions of the Earth. In view of the very small magnitude of the moment of inertia of the inner core relative to that of the the Earth as a whole, one expects from physical considerations that inclusion of the inner core in the dynamics should lead to a new nutational normal mode with a natural frequency not too far from that of the free core nutation, and to an associated weak resonance in the amplitude of forced nutations. We present here a treatment of the nutation problem for an oceanless, elastic, spheroidally stratified Earth, with the dynamical role of the inner core explicitly included in the formulation. As a preliminary to the setting‐up of dynamical equations, we devote some attention to a careful definition of a suitable coordinate system and of certain basic dynamical variables. We use the approach of Sasao et al. (1980), with their system of dynamical equations enlarged by the inclusion of two additional equations which are needed to describe the rotational motion of the inner core. An extension and sharpening of a line of reasoning employed by them enables us to derive expressions for the torques which couple the mantle and the fluid outer core to the solid inner core. Solving the enlarged system of equations, we show that a new nearly diurnal eigenfrequency does emerge; a rough estimate places it not very far from the prograde annual tidal excitation frequency, so that possible resonance effects on nutation amplitudes need careful consideration. Another eigenfrequency, attributable to a wobble of the inner core, is also found; its value is estimated to be a few times smaller than the wobble frequency that the inner core would have in the absence of couplings to the rest of the Earth. Considering an expansion, in terms of resonance contributions, of the amplitude of forced nutations normalized relative to that for a corresponding rigid Earth model, we indicate how the coefficients in the expansion are related to those in expansions of the type used by Wahr (1981b). Finally, we discuss the problem of comparing observed nutation amplitudes with those computed on the basis of Earth models generated from seismological data, with special reference to the fact that the dynamical ellipticity of the Earth, as computed from published Earth models which assume the condition of hydrostatic equilibrium, differs significantly from that determined from the precession constant. Numerical results, corrections for unmodeled effects, and comparison with observational results will be dealt with in accompanying papers.

In this note we present a remarkable nonlinear system which has the property that all its bounded periodic motions are simple harmonic. The system is a particle obeying the highly nonlinear equation of motionwhence the canonical momentum p and the Hamiltonian II may be obtained:The Lagrangian (2) is the single particle analogue of the Lagrangian densityof a relativistic scalar field. Field systems with this kind of Lagrangian are of wide current interest [1,2] in the context of elementary particle theory, and have in fact been considered especially in the context of chiral Lagrangian theories of pion interactions. Here we limit ourselves to obtaining the solutions of Eq. (1), which immediately lead to the plane wave (c-number) solutions of the field equation of (5) where i(y) = f f(y) dy and the constant of integration has been written as -(«/X) for future convenience.

The presence of an internal magnetic field influences of the Earth's nutation through the effects of electromagnetic torques at the boundaries of the fluid core. We calculate the effect of electromagnetic torques on nutation by combining a solution for the full hydromagnetic response of the fluid core with the nutation theory of Mathews et al. [2002]. The coupling of the fluid outer core to the mantle and solid inner core is described by two complex constants, KCMB and KICB, that characterize the electromagnetic torques at the core‐mantle boundary (CMB) and the inner core boundary (ICB). Predictions for KCMB and KICB are compared with estimates inferred from observations of the Earth's nutation. The estimate of KCMB can be explained by the presence of a thin conducting layer at the base of the mantle with a total conductance of 108 S. The overall root‐mean‐square (RMS) radial field at the CMB is 0.69 mT, which is partitioned into a dipole component (0.264 mT) and a nondipole component (0.64 mT). (The latter is represented using a uniform radial field.) The estimate of KICB can be explained with a mixture of dipole and nondipole components. The overall RMS field at the ICB is 7.17 mT, though smaller values are inferred when small adjustments are made to the dynamic ellipticity of the inner core and/or the fluid density at the boundary. The minimum RMS radial field required to explain the nutation observations is 4.6 mT.