  In this paper we consider some aspects of the relativistic dynamics of a
cylindrical shell of counter rotating particles. In some sense these are the
simplest systems with a physically acceptable matter content that display in a
well defined sense an interaction with the radiative modes of the gravitational
field. These systems have been analyzed previously, but in most cases resorting
to approximations, or considering a particular form for the initial value data.
Here we show that there exists a family of solutions where the space time
inside the shell is flat and the equation of motion of the shell decouples
completely from the gravitational modes. The motion of the shell is governed by
an equation of the same form as that of a particle in a time independent one
dimensional potential. We find that under appropriate initial conditions one
can have collapsing, bounded periodic, and unbounded motions. We analyze and
solve also the linearized equations that describe the dynamics of the system
near a stable static solutions, keeping a regular interior. The surprising
result here is that the motion of the shell is completely determined by the
configuration of the radiative modes of the gravitational field. In particular,
there are oscillating solutions for any chosen period, in contrast with the
"approximately Newtonian plus small radiative corrections" motion expectation.
We comment on the physical meaning of these results and provide some explicit
examples. We also discuss the relation of our results to the initial value
problem for the linearized dynamics of the shell.
