2 KINDS OF GENERALIZED TAUB-NUT METRICS AND THE SYMMETRY OF ASSOCIATED DYNAMICAL-SYSTEMS

A number of researches have been made for the (Euclidean) Taub-NUT metric, because the geodesic for this metric describes approximately the motion of well separated monopole-monopole interaction. From the viewpoint of dynamical systems, It is well known also that the Taub-NUT metric admits the Kepler-type symmetry, and hence provides a non-trivial generalization of the Kepler problem. More specifically speaking, because of an U(1) symmetry, the geodesic flow system as a Hamiltonian system for the Taub-NUT metric is reduced to a Hamiltonian system which admits a conserved Runge-Lenz-like vector in addition to the angular momentum vector, and thereby whose trajectories turn out to be conic sections. In particular, all the bounded trajectories of the reduced system are closed. In this paper, the Taub-NUT metrics is generalized so that the reduced system may remain to have the property that all of bounded trajectories are closed. On the application of Bertrand's method to the reduced system, two types of systems are found; one is called the Kepler-type system and the other the harmonic oscillator-type system. Correspondingly, two types of metrics come out: the Kepler-type metric and the harmonic oscillator-type metric. Furthermore, the symmetry of the Kepler-type system and of the harmonic oscillator-type system are studied through forming accidental first integrals. Thus the generalization of the Taub-NUT metric accomplishes non-trivial generalizations of the Kepler problem and the harmonic oscillator