We study the motion of neutral and charged spinning bodies in curved space-time in the test-particle limit. We construct equations of motion using a closed covariant Poisson-Dirac bracket formulation which allows for different choices of the hamiltonian. We derive conditions for the existence of constants of motion and apply the formalism to the case of spherically symmetric space-times. We show that the periastron of a spinning body in a stable orbit in a Schwarzschild or Reissner-Nordstr{o}m background not only precesses, but also varies radially. By analysing the stability conditions for circular motion we find the innermost stable circular orbit (ISCO) as a function of spin. It turns out that there is an absolute lower limit on the ISCOs for increasing prograde spin. Finally we establish that the equations of motion can also be derived from the Einstein equations using an appropriate energy-momentum tensor for spinning particles.
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