Calculation of vortex states using dissipative structures with Dirac constraint theory

Dynamical systems, finite or infinite, that describe physical phenomena typically have parts that are in some sense Hamiltonian and parts that can be recognized as dissipative, with the Hamiltonian part being generated by a Poisson bracket and the dissipative part being some kind of gradient flow. The description of Hamiltonian systems has received much attention over nearly two centuries and, although some forms of dissipation have received general attention, the understanding and classification of dissipative dynamics is a much broader topic and consequently less well developed. Early formalisms for dissipation include that due to Rayleigh and the Cahn-Hilliard type of system, but formalisms of greater complexity and interest are those that emerge from a Hamiltonian structure. Examples of the latter include double bracket dynamics due to Brockett and Young et al. and metriplectic flows introduced in [1]. (See [2, 3] and references therein for general discussion.) Double bracket flows dissipate energy while preserving Casimir invariants, while metriplectic flows embody the first and second laws of thermodynamics and, thus, conserve energy and produce entropy. Both double bracket and metriplectic flows have interesting algebraic, geometric, and functional analytic properties (see [3]), depending on the context, but of interest here is how they can be used for practical computations. For illustration purposes, consider the simple case of a finite-dimensional manifold Zs, that is both symplectic and Reimannian. Because Zs is symplectic, given any smooth function f : Zs → R there naturally corresponds the Hamiltonian vector field Xf := [z, f ], where [ , ] is the Poisson bracket and z denotes coordinates of Zs. Because Zs is Riemannian it has a metric g(X,Y ) defined on vector fields X,Y . With this machinery there is a symmetric bracket on pairs of functions given in terms of two Hamiltonian vector fields