Orbital classification of geodesic flows on two-dimensional ellipsoids. The Jacobi problem is orbitally equivalent to the integrable Euler case in rigid body dynamics
It is well-known that the geodesic flow of the Riemannian metric induced on an ellipsoid in \(\mathbb{R}^3\) is a completely integrable Hamiltonian system \(v_J\) defined on the four-dimensional cotangent bundle \(T^*S^2\). Since the three-dimensional constant-energy surfaces are invariant under the flow, it suffices to investigate the topological orbital classification of such systems on these three-dimensional surfaces. In this paper it is shown that the geodesic flows on two ellipsoids are orbitally topologically equivalent if and only if the ellipsoids are similar (that is, their semiaxes are proportional). Another important integrable system is the so-called integrable Euler case in rigid body dynamics in \(\mathbb{R}^3\), \(v_E\) which describes the rotation of a rigid body fixed at its centre of mass. It is known that, after reduction, this system is also defined on a four-dimensional symplectic manifold (as a system with two degrees of freedom), which is diffeomorphic to \(T^*S^2\). Thus both systems \(v_J\) and \(v_E\) are actually defined on the same manifold \(T^*S^2\). Since on the zero section of \(T^* S^2\) there is no motion (the Hamiltonians have singularities), this section can be removed and both systems can be considered on the four-manifold \(\widetilde M=T^* S^2 \backslash\)\{zero section\}. It is then proved that under an appropriate choice of semiaxes of an ellipsoid and moments of inertia of a rigid body, the geodesic flow on the ellipsoid, \(v_J\), is orbitally equivalent to the Euler system \(v_E\) (with zero area constant) on the manifold \(\widetilde M\). In addition, the explicit formulae that establish a one-to-one correspondence (up to similarity) between the squared semiaxes of the ellipsoid and the principal moments of inertia of the corresponding rigid body are given. The main results of this paper follow from the authors' theory of orbital classification of nondegenerate integrable systems with two degrees of freedom on three-dimensional energy levels [Russ. Acad. Sci., Sb., Math. 81, No. 2, 421-465 (1995); translation from Mat. Sb. 185, No. 4, 27-80 (1994; Zbl 0828.58020); Russ. Acad. Sci., Sb., Math. 82, No. 1, 21-63 (1995); translation from Mat. Sb. 185, No. 5, 27-78 (1994; Zbl 0868.58026)].
completely integrable Hamiltonian system
scientific article; zbMATH DE number 898970
scientific article