  The Hamiltonian formulation of a plasma four-field fluid model that describes
collisionless reconnection is presented. The formulation is noncanonical with a
corresponding Lie-Poisson bracket. The bracket is used to obtain new
independent families of invariants, so-called Casimir invariants, three of
which are directly related to Lagrangian invariants of the system. The Casimirs
are used to obtain a variational principle for equilibrium equations that
generalize the Grad-Shafranov equation to include flow. Dipole and homogeneous
equilibria are constructed. The linear dynamics of the latter is treated in
detail in a Hamiltonian context: canonically conjugate variables are obtained;
the dispersion relation is analyzed and exact thresholds for spectral stability
are obtained; the canonical transformation to normal form is described; an
unambiguous definition of negative energy modes is given; and thresholds
sufficient for energy-Casimir stability are obtained. The Hamiltonian
formulation also is used to obtain an expression for the collisionless
conductivity and it is further used to describe the linear growth and nonlinear
saturation of the collisionless tearing mode.
