A hierarchy of noncanonical Hamiltonian systems

The dynamics of an ideal fluid or plasma is constrained by topological invariants such as the circulation of (canonical) momentum. In the Hamiltonian formalism, topological invariants restrict the orbits to submanifolds of the phase space. While the coadjoint orbits have a natural symplectic structure, the global geometry of the degenerate (constrained) Poisson manifold can be very complex. Some invariants are represented by the center of the Poisson algebra (i.e., the Casimir elements such as the helicities), and then, the global structure of phase space is delineated by Casimir leaves. However, a general constraint is not necessarily integrable, which precludes the existence of an appropriate Casimir element; the circulation is an example of such an invariant. In this work, we formulate a systematic method to embed a Hamiltonian system in an extended phase space; we introduce mock fields and extend the Poisson algebra. A mock field defines a new Casimir element, a cross helicity, which represents topological constraints which are not integrable in the original phase space. Changing the perspective, a singularity of the extended system may be viewed as a subsystem on which the mock fields (though they are actual fields, when viewed from the extended system) vanishes, i.e., the original system. This hierarchical relation of degenerate Poisson manifolds enables us to see the “interior” of a singularity as a sub Poisson manifold. The theory can be applied to describe the bifurcation and instabilities in a wide class of general Hamiltonian systems [Yoshida & Morrison, Fluid Dyn. Res. 46 (2014), 031412],