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 met ric, because the geodesic for this metric describes approximately the motion of well separated monopole-monopole interaction. From the viewp oint of dynamical systems, It is well known also that the Taub-NUT met ric admits the Kepler-type symmetry, and hence provides a non-trivial generalization of the Kepler problem. More specifically speaking, beca use of an U(1) symmetry, the geodesic flow system as a Hamiltonian sys tem for the Taub-NUT metric is reduced to a Hamiltonian system which a dmits a conserved Runge-Lenz-like vector in addition to the angular mo mentum vector, and thereby whose trajectories turn out to be conic sec tions. In particular, all the bounded trajectories of the reduced syst em are closed. In this paper, the Taub-NUT metrics is generalized so t hat the reduced system may remain to have the property that all of bou nded trajectories are closed. On the application of Bertrand's method to the reduced system, two types of systems are found; one is called t he Kepler-type system and the other the harmonic oscillator-type syste m. Correspondingly, two types of metrics come out: the Kepler-type met ric 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 gener alization of the Taub-NUT metric accomplishes non-trivial generalizati ons of the Kepler problem and the harmonic oscillator