Singular Casimir elements: their mathematical justification and physical implications

摘要

Bifurcation of equilibrium points in fluids or plasmas is studied using the notion of Casimir foliation that occurs in the noncanonical 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.