Attractor-driven matter

Abstract

The state of a classical point-particle system may often be specified by giving the position and momentum for each constituent particle. For non-pointlike particles, the center-of-mass position may be augmented by an additional coordinate that specifies the internal state of each particle. The internal state space is typically topologically simple, in the sense that the particle’s internal coordinate belongs to a suitable symmetry group. In this paper, we explore the idea of giving internal complexity to the particles, by attributing to each particle an internal state space that is represented by a point on a strange (or otherwise) attracting set. It is, of course, very well known that strange attractors arise in a variety of nonlinear dynamical systems. However, rather than considering strange attractors as emerging from complex dynamics, we may employ strange attractors to drive such dynamics. In particular, by using an attractor (strange or otherwise) to model each particle’s internal state space, we present a class of matter coined “attractor-driven matter.” We outline the general formalism for attractor-driven matter and explore several specific examples, some of which are reminiscent of active matter. Beyond the examples studied in this paper, our formalism for attractor-driven dynamics may be applicable more broadly, to model complex dynamical and emergent behaviors in a variety of contexts.

Rendezvous control for a double-integrator multi-particle system on Lorentz group SO (1,1)

“…From equations ( 29), (30), and (32), we find that the closed-loop system (13) and ( 15) is equivalent to…”

“…25,26 The Lorentz group is a Lie group of symmetries of the Minkowski spacetime in special relativity. Many physics laws, equations, and theories, e.g., Maxwell's field equations in the theory of electromagnetism, 27,28 the Dirac equation in the theory of the electron, 29,30 and the Standard Model of particle physics, 31,32 respect the Lorentz symmetry. Besides, in engineering, the Lorentz group is also used to characterize the catadioptric fundamental matrices that are applied to study the catadioptric cameras.…”