Degenerate dynamical systems

Abstract

Dynamical systems whose symplectic structure degenerates, becoming noninvertible at some points along the orbits are analyzed. It is shown that for systems with a finite number of degrees of freedom, like in classical mechanics, the degeneracy occurs on domain walls that divide phase space into nonoverlapping regions each one describing a nondegenerate system, causally disconnected from each other. These surfaces are characterized by the sign of the Liouville's flux density on them, behaving as sources or sinks of orbits. In this latter case, once the system reaches the domain wall, it acquires a new gauge invariance and one degree of freedom is dynamically frozen, while the remaining degrees of freedom evolve regularly thereafter.

General relativity from a gauged Wess-Zumino-Witten term

“…Generically, as it happens with all higher dimensional Chern-Simons theories, the system will possess degenerate dynamical sectors [26,27,28]. A deeper understanding of this problem would be required prior to any study of the quantum properties of the action (9).…”

Black hole solutions in Chern-Simons AdS supergravity

“…At those degeneracy surfaces the system acquires extra gauge symmetry and looses dynamical degrees of freedom. This is a generic feature of higher dimensional CS systems [8,9,14], but it has been known to exist in all generic Lovelock theories [27][28][29] (see also the discussions in [30][31][32] and references therein), as well as in many mechanical systems [33]. In the above solution, both χ r (r) and ψ r (r) remain undetermined, as θ(r) and η(r) are arbitrary functions of r. General Lovelock theory has a pathological structure of its phase space because of the non-invertible relation between the metric and its conjugate momentum [28,29].…”

Dynamical sectors of a relativistic two particle model

“…It seems that from the point of view of the true degrees of freedom the evolution among sectors does not matter since the spacetime variables are unaffected by this phenomenon. Evolution among sectors occurs in the model studied in [11]. 4 …”

“…1 In the case of four dimensions, bigravity contains also sectors; see for example [8]. For other models with sectors see [9][10][11][12].…”