Topological classification of integrable Hamiltonian systems in a potential field on surfaces of revolution (Q2817569)

The author considers a class of mechanical systems that describe the motion of a point mass on a two-sphere with a ``metric of revolution''. This metric is given by a function \(f(r)\) in arbitrary smooth potential field \(V(r)\). When the surface of revolution can be embedded in \(\mathbb{R}^3\), the function \(f(r)\) is the generator of the surface of revolution.NEWLINENEWLINE Such systems are known to be complete and Liouville integrable, so the topological classification theory of Fomenko et al. can be used to analyze them. In this paper, the author gives a topological characterization of such surface of revolution systems. Fomenko's theory provides an effective means of assigning a discrete invariant. This invariant gives a complete description of the Liouville foliation for the system in question (up to leaf-wise equivalence). The invariant has the structure of a graph with numerical marks and is called the ``marked molecule'' or Fomenko-Zieschang invariant.NEWLINENEWLINE The author's main result is that integrable systems with a smooth potential on two-dimensional surfaces of revolution, when restricted to a connected component of a three-dimensional isoenergy surface, are topologically equivalent to certain classical integrable dynamical systems.