Kendall's problem on a sphere (Q1936526)

The present paper considers Kendall's problem on a sphere. A sphere \(\Sigma \) of radius 1, which rolls without slipping and without twisting over a sphere \(M\) of radius \(R > 1\) is considered with a fixed orthonormal frame \(\mathcal R\) attached to \(\Sigma\). Let be given two points \(p\) and \(q\) on M (not necessarily different) and two states of \(\mathcal R\): \({\mathcal R}_p\) and \({\mathcal R}_q \) (also not necessarily different). It is supposed that \(\Sigma \) is on \(p\) with the frame \(\mathcal R\) in the position \({\mathcal R}_p\). The problem is to find the number of displacements moving \(\Sigma\) without slipping and without twisting over geodesics of \(M\) to the point \(q\) where the frame \(\mathcal R\) must arrive in the state \({\mathcal R}_q \). It is proved, that it is possible in a sequence of no more than four movements.