  We consider different spin supplementary conditions (SSC) for a spinning
compact binary with the leading-order spin-orbit (SO) interaction. The
Lagrangian of the binary system can be constructed but it is
acceleration-dependent in two cases of SSC. We rewrite the generalized
Hamiltonian formalism proposed by Ostrogradsky and compute the conserved
quantities and the dissipative part of relative motion during the gravitational
radiation of each SSC. We give the orbital elements and observed quantities of
the SO dynamics, for instance the energy and the orbital angular momentum
losses and waveforms and discuss their SSC dependence.
