**DDA2001, April2001**

*Session 2. Very Big Things*

Monday, 1:30-3:40pm
[Previous] |
[Session 2] |
[Next]

## [2.03] Integrals of Motion in Kuzmin-Like Galaxy Potentials

*J. E. Tohline, K. Voyages (LSU)*

We have identified an orthogonal curvilinear coordinate
system (\xi_{1}, \xi_{2}, \xi_{3}) for which surfaces of
constant \xi_{1} exactly coincide with equipotential
surfaces in the Kuzmin galaxy potential. The three principal
coordinates are: \xi_{1} \equiv [x^{2} + y^{2} + (a + |z|)^{2}
]^{1/2}, \xi_{2} \equiv \arctan[ (|z|/z) (x^{2} + y^{2})^{1/2} /
(a + |z|)], \xi_{3} \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 \xi_{1} will exhibit the same number of
integrals of motion as does a spherically symmetric
potential. In such Kuzmin-like potentials, the vector **
J** \equiv **e**_{1} \xi_{1} **\times v** is the analog of
the specific angular momentum vector in spherically
symmetric potentials. Furthermore, in potentials of the form
\Phi \propto 1/\xi_{1}, all three Cartesian components of
the vector **L** \equiv [**v \times J** + **e**_{1}
(\xi_{1} \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.

[Previous] |
[Session 2] |
[Next]