Degenerate Dynamical Systems

TL;DR

This paper investigates dynamical systems with degenerate symplectic structures, revealing how phase space is partitioned into regions separated by domain walls that influence system behavior and gauge invariance.

Contribution

It introduces a framework for analyzing degenerate symplectic systems, showing how degeneracy leads to phase space division and changes in gauge invariance.

Findings

Degeneracy occurs on domain walls dividing phase space.

Domain walls act as sources or sinks of orbits.

Reaching a domain wall can freeze a degree of freedom.

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.