Geodesic flow on an intersection of several confocal quadrics in $\mathbb{R}^n$

Abstract

By the Jacobi-Chasles theorem, for each geodesic on an $n$-axial ellipsoid in $n$-dimensional Euclidean space, apart from it there exist also $n-2$ quadrics confocal with it that are tangent to all the tangent lines of this geodesic. It is shown that this result also holds for the geodesic flow on the intersection of several nondegenerate confocal quadrics. As in the case of the Jacobi-Chasles theorem, this fact ensures the integrability of the corresponding geodesic flow. For each compact intersection of several nondegenerate confocal quadrics its homeomorphism class is determined, and it turns out that such an intersection is always homeomorphic to a product of several spheres. Also, a sufficient condition for a potential is presented which ensures that the addition of this potential preserved the integrability of the corresponding dynamical system on the intersection of an arbitrary number of confocal quadrics. Bibliography: 16 titles.