  Topological constraints play a key role in the self-organizing processes that
create structures in macro systems. In fact, if all possible degrees of freedom
are actualized on equal footing without constraint, the state of
"equipartition" may bear no specific structure. Fluid turbulence is a typical
example - while turbulent mixing seems to increase entropy, a variety of
sustained vortical structures can emerge. In Hamiltonian formalism, some
topological constraints are represented by Casimir invariants (for example,
helicities of a fluid or a plasma), and then, the effective phase space is
reduced to the Casimir leaves. However, a general constraint is not necessarily
integrable, which precludes the existence of an appropriate Casimir invariant;
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 phantom fields and extend the Poisson algebra. A phantom field
defines a new Casimir invariant, a cross helicity, which represents a
topological constraint that is 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 phantom fields (though they are actual fields, when
viewed from the extended system) vanish, 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 bifurcations and instabilities in a wide class of general
Hamiltonian systems [Yoshida & Morrison, Fluid Dyn. Res. 46 (2014), 031412].
