Dissipative Perturbations of 3d Hamiltonian Systems

In this article we present some results concerning natural dissipative perturbations of 3d Hamiltonian systems. Given a Hamiltonian system ẋ = PdH, and a Casimir function S, we construct a symmetric covariant tensor g, so that the modified (so-called “metriplectic”) system ẋ = PdH + gdS satisfies the following conditions: dH is a null vector for g, and dS(gdS) ≤ 0. Along solutions to a dynamical system of this type, the Hamiltonian function H is preserved, while the function S decreases, i.e. S is dissipated by the system. We are motivated by the example of a “relaxing rigid body” by Morrison [7] in which systems of this type were introduced. Introduction In his article “A Paradigm for Joined Hamiltonian and Dissipative Systems” (see [7]), P.J. Morrison introduced a natural geometric formulation of dynamical systems that exhibit both conservative and nonconservative characteristics. Historically, conservative systems have been modelled geometrically as Hamiltonian systems of the form ẋ = PdH where P is a Poisson tensor and H is a smooth function (Hamiltonian function). In a Hamiltonian system, the equality dH/dt = dHPdH = 0 can be interpreted as conservation of the “energy” H ; thus such systems are natural models for conservative dynamics (see [1]). Nonconservative or dissipative systems can also be described geometrically as gradient systems: ẋ = gdS where g is a symmetric tensor, S is a smooth function, and dS/dt = dSgdS is typically negative definite. Since the function S is not conserved, it may be interpreted as a form of energy that is dissipated (or as the negative of “entropy” which is produced by the system) (see [9]). Morrison’s formulation combines these two types of systems into a so-called Metriplectic System ẋ = PdH + gdS, with the additional requirements that H remains a conserved quantity and S continues to be dissipated. These requirements can be met if the following conditions on H and S are satisfied PdS = gdH = 0. That is, S is a Casimir function for the Poisson tensor P and dH is a null vector for the symmetric tensor g. Morrison applied this formulation to the equations for the rigid-body with dissipation and the Vlasov-Poisson equations for plasma with collisions. The formulation of dissipative systems as combined Hamiltonian and gradient systems has also been studied by [2], [3], [6],[5], [10] and others in mathematical physics and control theory. 1