Hamiltonian equations for multiple-collective-variable theories of nonlinear Klein-Gordon equations: A projection-operator approach

Abstract

We consider classical Hamiltonian systems in which there exist collective modes where the motion associated with each collective mode is describable by a collective coordinate. The formalism we develop is applicable to both continuous and discrete systems where the aim is to investigate the dynamics of kink or solitonlike solutions to nonlinear Klein-Gordon equations which arise in field theory and condensed-matter theory. We present a new calculational procedure for obtaining the equations of motion for the collective coordinates and coupled fields based on Dirac's treatment of constrained Hamiltonian systems. The virtue of this new (projection-operator) procedure is the ease with which the equations of motion for the collective variables and coupled fields are derived relative to the amount of work needed to calculate them from the Dirac brackets directly. Introducing collective coordinates as dynamical variables into a system enlarges the phase space accessible to the possible trajectories describing the system's evolution. This introduces extra solutions to the new equations of motion which do not satisfy the original equations of motion. It is therefore necessary to introduce constraints in order to conserve the number of degrees of freedom of the original system. We show that the constraints have the effect of projecting out the motion in the enlarged phase space onto the appropriate submanifold corresponding to the available phase space of the original system. We show that the Dirac bracket accomplishes this projection, and we give an explicit formula for this projection operator. We use the Dirac brackets to construct a family of canonical transformations to the system of new coordinates (which contains the collective variables) and to construct a Hamiltonian in this new system of variables. We show the equations of motion that are derived through the lengthy Dirac bracket prescription are obtainable through the simple projection-operator procedure. We provide examples that illustrate the ease of this projection-operator method for the single- and multiple-collective-variable cases. We also discuss advantages of particular forms of the Ansatz used for introducing the collective variables into the original system.