Spontaneous symmetry breaking and neutral stability in the noncanonical Hamiltonian formalism

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Abstract
The noncanonical Hamiltonian formalism is based upon a generalization of the Poisson bracket, a particular form of which is possessed by continuous media fields. Associated with this generalization are special constants of motion called Casimir invariants. These are constants that can be viewed as being built into the phase space, for they are invariant for all Hamiltonians. Casimir invariants are important because when added to the Hamiltonian they yield an effective Hamiltonian that produces equilibrium states upon variation. The stability of these states can be ascertained by a second variation. Goldstone’s theorem, in its usual context, determines zero eigenvalues of the mass matrix for a given vacuum state, the equilibrium with minimum energy. Here, since for fluids and plasmas the vacuum state is uninteresting, we examine symmetry breaking for general equilibria. Broken symmetries imply directions of neutral stability. Two examples are presented: the nonlinear Alfvén wave of plasma physics and the Korteweg–de Vries soliton.

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