| Abstract: | It is usually believed that we know everything to be known for any separable Hamiltonian system, i.e. an integrable system
in which we can separate the variables in some coordinate system (e.g. see Lichtenberg and Lieberman 1992, Regular and Chaotic Dynamics, Springer). However this is not always true, since through the separation the solutions may be found only up to quadratures,
a form that might not be particularly useful. A good example is the two-fixed-centers problem. Although its integrability
was discovered by Euler in the 18th century, the problem was far from being considered as completely understood. This apparent
contradiction stems from the fact that the solutions of the equations of motion in the confocal ellipsoidal coordinates, in
which the variables separate, are written in terms of elliptic integrals, so that their properties are not obvious at first
sight. In this paper we classify the trajectories according to an exhaustive scheme, comprising both periodic and quasi-periodic
ones. We identify the collision orbits (both direct and asymptotic) and find that collision orbits are of complete measure
in a 3-D submanifold of the phase space while asymptotically collision orbits are of complete measure in the 4-D phase space.
We use a transformation, which regularizes the close approaches and, therefore, enables the numerical integration of collision
trajectories (both direct and asymptotic). Finally we give the ratio of oscillation period along the two axes (the ‘rotation
number’) as a function of the two integrals of motion.
This revised version was published online in July 2006 with corrections to the Cover Date. |
