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We derive the equations of motion for an $N$-body system in theEinstein-Cartan gravity theory at the first post-Newtonian order by exploitingthe Weyssenhoff fluid as the spin model. Our approach consists in performingthe point-particle limit of the continuous description of the gravitationalsource. The final equations provide a hint for the validity of the effacingprinciple at 1PN level in Einstein-Cartan model. The analogies with the generalrelativistic dynamics involving the macroscopic angular momentum are alsodiscussed.

First post-Newtonian $N$-body problem in Einstein-Cartan theory with the Weyssenhoff fluid: equations of motion