Mechanics of incompressible test bodies moving on λ‐spheres

Abstract

The considerations of the mechanics of incompressible test bodies moving on spheres are generalized to the case of λ‐spheres. Both surfaces are examples of Riemannian manifolds realized as two‐dimensional manifolds with strictly positive Gaussian curvature under the condition λ < 1/3 for the λ‐spheres. A convenient parametrization of λ‐spheres embedded as two‐dimensional surfaces of revolution into the three‐dimensional Euclidean space is derived. The so‐obtained parametrization is expressed in a concise form via the elliptic integrals of the first, second, and third kind. Next, the geodetic motion is considered. The explicit solutions for two branches of the incompressible motion are obtained in the parametric form. For the special case of geodetics corresponding to meridians taken as geodesics their analysis is performed in detail. In the latter case, the so‐obtained geodetic solutions are reduced to the incomplete elliptic integrals of the first, second, and third kind.