Regular and chaotic dynamics of non-spherical bodies. Zeldovich's
  pancakes and emission of very long gravitational waves
  In this paper we review a recently developed approximate method for
investigation of dynamics of compressible ellipsoidal figures. Collapse and
subsequent behaviour are described by a system of ordinary differential
equations for time evolution of semi-axes of a uniformly rotating, three-axis,
uniform-density ellipsoid. First, we apply this approach to investigate dynamic
stability of non-spherical bodies. We solve the equations that describe, in a
simplified way, the Newtonian dynamics of a self-gravitating non-rotating
spheroidal body. We find that, after loss of stability, a contraction to a
singularity occurs only in a pure spherical collapse, and deviations from
spherical symmetry prevent the contraction to the singularity through a
stabilizing action of nonlinear non-spherical oscillations. The development of
instability leads to the formation of a regularly or chaotically oscillating
body, in which dynamical motion prevents the formation of the singularity. We
find regions of chaotic and regular pulsations by constructing a Poincare
diagram. A real collapse occurs after damping of the oscillations because of
energy losses, shock wave formation or viscosity. We use our approach to
investigate approximately the first stages of collapse during the large scale
structure formation. The theory of this process started from ideas of Ya. B.
Zeldovich, concerning the formation of strongly non-spherical structures during
nonlinear stages of the development of gravitational instability, known as
'Zeldovich's pancakes'. In this paper the collapse of non-collisional dark
matter and the formation of pancake structures are investigated approximately.
We estimate an emission of very long gravitational waves during the collapse,
and discuss the possibility of gravitational lensing and polarization of the
cosmic microwave background by these waves.
