  We have computed numerically the Casimir force between two identical pistons
inside a very long cylinder, considering different shapes for the pistons. The
pistons can be considered as quantum billiards, whose spectrum determines the
vacuum force. The smooth part of the spectrum fixes the force at short
distances, and depends only on geometric quantities like the area or perimeter
of the piston. However, correcting terms to the force, coming from the
oscillating part of the spectrum which is related to the classical dynamics of
the billiard, are qualitatively different for classically integrable or chaotic
systems. We have performed a detailed numerical analysis of the corresponding
Casimir force for pistons with regular and chaotic classical dynamics. For a
family of stadium billiards, we have found that the correcting part of the
Casimir force presents a sudden change in the transition from regular to
chaotic geometries.
