Mechanics of incompressible test bodies moving in Riemannian spaces

In the present paper, we have discussed the mechanics of incompressible test bodies moving in Riemannian spaces with nontrivial curvature tensors. For Hamilton's equations of motion, the solutions have been obtained in the parametrical form and the special case of the purely gyroscopic motion on the sphere has been discussed. For the geodetic case when the potential is equal to zero, the comparison between the geodetic and geodesic solutions has been done and illustrated in details for the case of a particular choice of the constants of motion of the problem. The obtained results could be applied, among others, in geophysical problems, for example, for description of the movement of continental plates or the motion of a drop of fat or a spot of oil on the surface of the ocean (e.g., produced during some “ecological disaster”), or generally in biomechanical problems, for example, for description of the motion of objects with internal structure on different curved two‐dimensional surfaces (e.g., transport of proteins along the curved biological membranes).