## Prof. U. Frauenfelder (Universität Augsburg)

### Families of periodic orbits in the restricted three body problem
and Arnold's J⁺ invariant

**Abstract:** This is joint work with Kai Cieliebak and Otto van Koert.
The restricted three body Problem describes the movement of a massless particle attracted by
two masses according to Newton's law of Gravitation. For example one could imagine a satellite
attracted by the earth and the moon, the moon attracted by the earth and the sun, or a planet
in a double star System. The trajectory of the massless particle is usually immersed except for
collisions with one of the masses or a phenomenon which is referred to by Hill as a moon of
maximal lunarity. By the theorem of Whitney-Graustein the rotation number is a complete invariant
for immersed loops in the plane up to homotopy. In a generic homotopy three disasters can occur -
triple intersection, inverse and direct self tangencies. Arnold's J⁺ invariant is
unchanged under the first two disasters but is sensible to direct self tangencies. For
families of periodic orbits in the restricted three body problems two additional disasters
can occur - occurence of cusps in the case of a moon of maximal lunarity and collisions.
We show how the theory of Arnold's J⁺ invariant can be modified to obtain invariants
for families of periodic orbits in the restricted three body problem.