Singular Casimir Elements: Their Mathematical Justification and Physical Implications

Bifurcation of equilibrium points in fluids or plasmas is studied using the notion of Casimir foliation that occurs in the noncanon- ical Hamiltonian formalism of the ideal dynamics. The nonlinearity of the system makes the Poisson operator inhomogeneous on phase space (the function space of the state variable), and creates a singularity where the nullity of the Poisson operator changes. The problem is an infinite-dimensional generalization of the theory of singular differential equations. Singular Casimir elements stemming from this singularity are unearthed using a generalization of the functional derivative that occurs in the Poisson bracket.