H. Varvoglis
Section of Astrophysics, Astronomy and Mechanics - University of Thessaloniki, Greece
The study of the motion of a point mass moving in the gravitational field
of two fixed attracting centers is a problem proposed and solved in the
18th century by Euler, as an intermediate step towards the solution of the
three-body problem. Shortly after, Lagrange showed that the
corresponding potential was separable in elliptical confocal co-ordinates
and later on, in 1949, Erikson and Hill found the corresponding third
integral of motion. Since then the problem has been considered as a non-
exciting example of a separable potential and it is included, as such, in
many textbooks of Theoretical Mechanics. We show that the two fixed
centers problem possesses two very interesting features, which put it in
the class of interesting dynamical systems:
(a) it possesses non-isolated periodic trajectories with characteristic
exponents equal to 0, a case mentioned in textbooks but for which
very few examples exist and
(b) it is a dynamical system with collision orbits which form a set of
complete measure: all numerically integrated trajectories that either
encircle only one of the centers (satellite orbits) or encircle both of
them but in alternating directions (lemniscates) are collision orbits.