  Recently we derived the next-to-next-to-leading order post-Newtonian
Hamiltonians at spin-orbit and spin(1)-spin(2) level for a binary system of
compact objects. In this talk the derivation of them will be shortly outlined
at an introductory level. We will also discuss some checks of our (complicated
and long) results in the first part of the talk. In the second part we will
show how to apply our results to the calculation of the last stable circular
orbit of such a binary system of black holes or neutron stars.
