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In general relativity, systems of spinning classical particles areimplemented into the canonical formalism of Arnowitt, Deser, and Misner [1].The implementation is made with the aid of a symmetric stress-energy tensor andnot a 4-dimensional covariant action functional. The formalism is valid toterms linear in the single spin variables and up to and including thenext-to-leading order approximation in the gravitational spin-interaction part.The field-source terms for the spinning particles occurring in the Hamiltonianare obtained from their expressions in Minkowski space with canonical variablesthrough 3-dimensional covariant generalizations as well as from a suitableshift of projections of the curved spacetime stress-energy tensor originallygiven within covariant spin supplementary conditions. The applied coordinateconditions are the generalized isotropic ones introduced by Arnowitt, Deser,and Misner. As applications, the Hamiltonian of two spinning compact bodieswith next-to-leading order gravitational spin-orbit coupling, recently obtainedby Damour, Jaranowski, and Schaefer [2], is rederived and the derivation of thenext-to-leading order gravitational spin(1)-spin(2) Hamiltonian, shown for thefirst time in [3], is presented.

ADM canonical formalism for gravitating spinning objects