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 dx/dt = PdH, and a Casimir function S, we construct a symmetric covariant tensor g, so that the modified (so-called 'metriplectic') system dx/dt = 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 P.J. Morrison in which systems of this type were introduced.