In general relativity, systems of spinning classical particles are
implemented into the canonical formalism of Arnowitt, Deser, and Misner [1].
The implementation is made with the aid of a symmetric stress-energy tensor and
not a 4-dimensional covariant action functional. The formalism is valid to
terms linear in the single spin variables and up to and including the
next-to-leading order approximation in the gravitational spin-interaction part.
The field-source terms for the spinning particles occurring in the Hamiltonian
are obtained from their expressions in Minkowski space with canonical variables
through 3-dimensional covariant generalizations as well as from a suitable
shift of projections of the curved spacetime stress-energy tensor originally
given within covariant spin supplementary conditions. The applied coordinate
conditions are the generalized isotropic ones introduced by Arnowitt, Deser,
and Misner. As applications, the Hamiltonian of two spinning compact bodies
with next-to-leading order gravitational spin-orbit coupling, recently obtained
by Damour, Jaranowski, and Schaefer [2], is rederived and the derivation of the
next-to-leading order gravitational spin(1)-spin(2) Hamiltonian, shown for the
first time in [3], is presented.Comment: REVTeX4, 18 pages. v1: published version. v2: corrected misprints in
(8.4) and (9.3), updated reference