Abstract
Metriplectic dynamics couple a Poisson bracket of the Hamiltonian description with a kind of metric bracket, 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 all fit within this framework. In this paper an application of metriplectic dynamics is presented that is of interest for the theory of control: a suitably chosen torque, expressed through a metriplectic extension of its “natural” Poisson algebra, an algebra obtained by reduction of a canonical Hamiltonian system, is applied to a free rigid body. On a practical ground, 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 provides a class of nonHamiltonian torques that can be added to the canonical Hamiltonian description of the free rigid body and reduce to metriplectic dissipation. In the canonical description these torques provide convergence to a higher dimensional attractor. The method of construction of such torques can be extended to other dynamical systems describing “machines” with non-Hamiltonian motion having attractors.
Highlights
This work is about a new feature of metriplectic dynamics [Morrison, 1986], an extension of the Hamiltonian formalism with dissipation that induces timeasymptotic convergence to equilibrium solutions
We focus on a particular finite-dimensional case, already introduced in [Morrison, 1986], in which the Hamiltonian system representing a free rigid body is perturbed with an external torque ⃗τservo suitably designed to modify the angular momentum L⃗ without changing the energy of the system
The metriplectic dynamics of [Morrison, 1986] in terms of ω⃗ relaxes to an asymptotic state of free rotation around a stable principal axis and has a point attractor
Summary
This work is about a new feature of metriplectic dynamics [Morrison, 1986], an extension of the Hamiltonian formalism with dissipation that induces timeasymptotic convergence to equilibrium solutions. In these sections it is shown that the rotation around a principal axis of inertia corresponds to an asymptotic equilibrium point in the ω⃗ space, while it is an attracting cylinder of periodic orbits in the canonical variables (χ, p⃗). An explicit finite-dimensional example, a set of ordinary differential equations for the free rigid body, will be presented This example illustrates the main point of the paper: how to amend a canonical Hamiltonian system in such a way that that it possesses relaxation to a limit cycle
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