  We solve the geodesic deviation equations for the orbital motions in the
Schwarzschild metric which are close to a circular orbit. It turns out that in
this particular case the equations reduce to a linear system, which after
diagonalization describes just a collection of harmonic oscillators, with two
characteristic frequencies. The new geodesic obtained by adding this solution
to the circular one, describes not only the linear approximation of Kepler's
laws, but gives also the right value of the perihelion advance (in the limit of
almost circular orbits). We derive also the equations for higher-order
deviations and show how these equations lead to better approximations,
including the non-linear effects. The approximate orbital solutions are then
inserted into the quadrupole formula to estimate the gravitational radiation
from non-circular orbits.
