The mechanics of inertial motion on the earth and on a rotating sphere

The Hamiltonian of the horizontal motion on the earth in the absence of friction and pressure gradient force is introduced as a modification of the Hamiltonian of the corresponding motion on a rotating sphere. This approach affords a clear identification of the dynamical effects of the earth’s rotation (i.e. Coriolis and centrifugal forces) versus those associated with horizontal component of gravity due to the eccentricity of the earth surface. Using the methods of analytical mechanics on manifolds, we find the exact solution of this 2-degrees-of-freedom integrable system. These considerations permit, in addition, a decomposition of the general trajectory into a motion (oscillation or rotation) along a great circle and the rotation of this great circle relative to the sphere or the earth. In the case of a rotating sphere, the particle rotates along the great circle with constant velocity independent of the great circle’s rotation relative to the sphere. In contrast, on the earth the velocity of the particle along the great circle (which can be either rotation or oscillation) varies with time and is coupled with the rotation of the great circle relative to the earth. We also show that on the earth the motion along the great circle can easily be reduced to the classical problem of simple pendulum. The geophysical implication of our results is that the mid-latitude inertial oscillations on the earth — oscillations that remain indefinitely in one hemisphere — originate from the earth’s minute eccentricity since they do not exist on a rotating sphere.