A collective-variable theory for nonlinear coherent excitations in classical Hamiltonian systems

We derive a collective-variable theory for nonlinear coherent excitations that is based on Dirac’s formalism of constrained Hamiltonian systems. No approximations are involved in this theory in the sense that we are able to formulate equations of motion for a new set of variables, including the collective variables themselves, which are equivalent to the original set of Hamilton’s equation of motion. We show that the dynamics of collective excitations has to be classified according to the rank of the so-called gyromatrix . Two extreme cases are particularly important: in the first case (zero gyromatrix) collective excitations behave very much like Newtonian particles of classical mechanics, while in the second case (regular gyromatrix) collective excitations are similar to charged, massless particles in an external magnetic field.