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J. E. Tohline, K. Voyages (LSU)
We have identified an orthogonal curvilinear coordinate system (\xi1, \xi2, \xi3) for which surfaces of constant \xi1 exactly coincide with equipotential surfaces in the Kuzmin galaxy potential. The three principal coordinates are: \xi1 \equiv [x2 + y2 + (a + |z|)2 ]1/2, \xi2 \equiv \arctan[ (|z|/z) (x2 + y2)1/2 / (a + |z|)], \xi3 \equiv \arctan[ (y/x) ]. When the equation of motion is written in terms of this ``Kuzmin'' coordinate system, the integrals of motion for particles moving in Kuzmin-like potentials are easily recognized. Specifically, any time-independent potential \Phi that is a function only of \xi1 will exhibit the same number of integrals of motion as does a spherically symmetric potential. In such Kuzmin-like potentials, the vector J \equiv e1 \xi1 \times v is the analog of the specific angular momentum vector in spherically symmetric potentials. Furthermore, in potentials of the form \Phi \propto 1/\xi1, all three Cartesian components of the vector L \equiv [v \times J + e1 (\xi1 \Phi)] --- an analog of the Laplace-Runge-Lenz vector --- also prove to be integrals of the motion. For two specific potentials --- the Kuzmin potential and a logarithmic Kuzmin-like potential --- we have derived analytical expressions defining the surfaces of section for particles in polar orbits, and in both cases have identified the domain of occupancy of box orbits, loop orbits, and periodic orbits. This work has been supported in part by the U.S. National Science Foundation through grant AST-9987344.