Usually, we call precession and nutation the motions of the Earth's rotation axis in the space reference frame. The polar motion (or "wobble") is the associated movement related to the Earth, modifying the latitude at a given place.
The nutations forced by the Moon, the Sun and the planets are generally much more important than the free nutations. Among these free oscillations, there is the Free Core Nutation (FCN) and the "Free Inner Core Nutation"(FICN).
The Free Core Nutation (FCN) is a mode related to non-alignment of the rotation axis of the core and of the mantle. It has a long period (of 432 days) in the celestial frame and is a retrograde mode. The physical parameters involved are the flattening of the Core Mantle Boundary (CMB) and the deformation of the CMB induced by the dynamic pressure. This mode is also called the Nearly Diurnal Free Wobble (NDFW, period of about 1 day), if observed in the terrestrial reference frame.
The lunar-solar attraction is responsible not only for the orbital motion, but also for the nearly periodic tidal motion of the Earth. The tidal force tends to deform the Earth to a prolate ellipsoid aligned with the Earth-Moon and Earth-Sun axis. The most obvious phenomenon, which represents the Earth's response to the luni-solar tidal force, is the ocean tide, but also there are deformations of the solid Earth, simply called Earth tides. There are additionally induced variations of the Earth's orientation in space (nutations). Observations of the Earth's responses to tidal forces supply important constraints to understand the Earth's interior. In particular, the variation of the observed response as a function of the frequency in the tidal band tells us something about the interior of our planet that seismology cannot provide. This variation with frequency is caused by resonant behaviour of the Earth near 1 cycle per day, due to a normal mode of the rotating Earth. This mode is the Nearly Diurnal Free Wobble (NDFW, period of about 1 day, period = 1 / [- (1 + (1 / 432))] day), if observed in the terrestrial reference frame, and the Free Core Nutation (FCN, period = 432 days), if referenced in the space frame.
Observing the NDFW / FCN is thus very useful to measure the CMB flattening and to obtain information about the dissipation effect at this interface. Fortunately, the eigenfrequency of the NDFW is located within the tidal band and is thus sensitive to the tidal forcing for the diurnal tides. In the space frame, the FCN is measured by the Very Long Baseline Interferometry (VLBI).
It is worthy to note that the observation of the NDFW / FCN is a non-seismic proof of the fluidity of the outer core.