Generalizations of classical Poisson brackets to include spin

The classical spin of a particle is essentially its angular momentum of rotation about an axis passing through its center of mass. The classical Poisson brackets are generalized to include spin for any system for which the rate of change of spin is given by $\frac{d\overrightarrow{s}}{\mathrm{dt}}=-\overrightarrow{s}\times \left(\frac{\partial H}{\partial \overrightarrow{s}}\right)$ where $H$ is the Hamiltonian. In a previous paper, we showed that this equation is satisfied by the Breit-Pauli Hamiltonian for a molecular system of electrons and nuclei interacting with an electromagnetic field. This Hamiltonian contains all of the fine-structure terms with the exception of the Lamb shift. Thus, our generalized Poisson brackets should apply to almost all molecular problems. Our generalized Poisson brackets are the same as those which Sudarshan and Mukunda derived by group-theoretic arguments for the direct product of a pure spin group with the group of transformations of generalized coordinates and their conjugate momenta.