Henk W. Broer, Heinz Hanßmann
A Galilean dance
(7392K, PostScript)
ABSTRACT. The four Galilean moons of Jupiter were discovered
by Galileo in the early 17th century, and their motion was first
seen as a miniature solar system. Around 1800 Laplace discovered
that the Galilean motion is subjected to an orbital 1:2:4-resonance
of the inner three moons Io, Europa and Ganymedes. In the early
20th century De Sitter gave a mathematical explanation for this
in a Newtonian framework. In fact, he found a family of stable
periodic solutions by using the seminal work of Poincar\'e, which
at the time was quite new. In this paper we review and summarize
recent results of Broer, Han{\ss}mann and Zhao on the motion
of the entire Galilean system, so including the fourth moon
Callisto. To this purpose we use a version of parametrised
Kolmogorov--Arnol'd--Moser theory where a family of multi-periodic
isotropic invariant three-dimensional tori is found that combines
the periodic motions of De Sitter and Callisto. The 3-tori are
normally elliptic and excite a family of invariant Lagrangean
8-tori that project down to librational motions. Both the 3- and
the 8-tori occur for an almost full Hausdorff measure set in the
product of corresponding dimension in phase space and a parameter
space, where the external parameters are given by the masses of
the moons.