Mechanics of incompressible test bodies moving on λ-spheres

The considerations of the mechanics of incompressible test bodies moving on spheres are generalized to the case of lambda-spheres. Both surfaces are examples of Riemannian manifolds realized as two-dimensional manifolds with strictly positive Gaussian curvature under the condition lambda < 1/3 for the lambda-spheres. A convenient parametrization of lambda-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 geodesics 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.

Automated summary

The mechanics of incompressible test bodies moving on spheres are generalized to lambda-spheres, with a derived parametrization and discussion of geodetic motion and specific geodesic solutions for meridians.