Title: Metriplectic formalism: friction and much more,
Abstract: The metriplectic formalism couples Poisson brackets of the Hamiltonian
description with metric brackets for describing systems with both Hamiltonian
and dissipative components. The construction builds in asymptotic convergence
to a preselected equilibrium state. Phenomena such as friction, electric
resistivity, thermal conductivity and collisions in kinetic theories are well
represented in this framework. In this paper we present an application of the
metriplectic formalism of interest for the theory of control: a suitable torque
is applied to a free rigid body, which is expressed through a metriplectic
extension of its "natural" Poisson algebra. On practical grounds, the effect is
to drive the body to align its angular velocity to rotation about a stable
principal axis of inertia, while conserving its kinetic energy in the process.
On theoretical grounds, this example shows how the non-Hamiltonian part of a
metriplectic system may include convergence to a limit cycle, the first example
of a non-zero dimensional attractor in this formalism. The method suggests a
way to extend metriplectic dynamics to systems with general attractors, e.g.
chaotic ones, with the hope of representing bio-physical, geophysical and
ecological models.