  We suggest to solve for the motion of the two body problem in General
Relativity by identifying the leading violation of conserved quantities,
referred to as (relativistic) anomalies, ordered by the post-Newtonian order at
which they appear. This differs from the standard procedure of obtaining the
full solution up to a prescribed order. We find that the reduced Hamiltonian
which describes the drift in the space of conserved quantities is given by the
average of the perturbation Hamiltonian. Using this approach the averaging is
done prior to the derivation of time evolution, thereby economizing the
computation. The computations become similar to those in the Hamilton-Jacobi
method, while staying in the more comfortable setting of the Hamiltonian
formulation. We apply this approach of leading anomalies and the drift
Hamiltonian to the binary problem and treat several perturbations: 1PN,
spin-orbit and spin-spin. On the way we discuss the interpretation of the
Laplace-Runge-Lenz vector as a generator of scale-preserving conformal
transformations in momentum space.
