The classical dynamic symmetry for the U(1)-Kepler problems

In this article we exhibit a Poisson realization of the simple real Lie algebra su(n,n) on the Poisson manifold (T∗X,{,}μ), i.e., a Lie algebra homomorphism from su(n,n) to C∞(T∗X,R),{,}μ. Consequently one obtains the Laplace-Runge-Lenz vector for the classical U(1)-Kepler problem of level n and magnetic charge μ. Since the McIntosh-Cisneros-Zwanziger-Kepler problems (MICZ-Kepler Problems) are the U(1)-Kepler problems of level 2, the work presented here is a direct generalization of the work by A. Barut and G. Bornzin (1971) on the classical dynamic symmetry for the MICZ-Kepler problems.