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As seen by an observer in the rotating frame, the earth’s small spheroidal deformations neutralize the centrifugal force, leaving only the smaller Coriolis force to govern the “inertial” motion of objects that move on its surface, assumed smooth and frictionless. Previous studies of inertial motion employ weakly spheroidal equations of motion that ignore the influence of the centrifugal force and yet treat the earth as a sphere. The latitude dependence of these equations renders them strongly nonlinear. We derive and justify these equations and use them to identify, classify, name, describe, and illustrate all possible classes of inertial motion, including a new class of motion called circumpolar waves, which encircle both poles during each cycle of the motion. We illustrate these classes using CorioVis, our freely available Coriolis visualization software. We identify a rotational/time-reversal symmetry for motion on the earth’s surface and use this symmetry to develop and validate closed-form small-amplitude approximations for the four main classes and one degenerate class of inertial motion. For these five classes, we supply calculations of experimentally relevant frequencies, zonal drifts, and latitude ranges. 

We analyze the rotational dynamics of six magnetic dipoles of identical strength at the vertices of a regular hexagon with a variable-strength dipole in the center. The seven dipoles spin freely about fixed axes that are perpendicular to the plane of the hexagon, with their dipole moments directed parallel to the plane. Equilibrium dipole orientations are calculated as a function of the relative strength of the central dipole. Small-amplitude perturbations about these equilibrium states are calculated in the absence of friction and are compared with analytical results in the limit of zero and infinite central dipole strength. Normal modes and frequencies are presented. Bifurcations are seen at two critical values of the central dipole strength, with bistability between these values.