that the field vanishes on the boundary of D) shows that the bracket obeys equation (\[INFCOSYM\]). Hence, in this case, we can use equation (\[INFNULL\]) to determine its Casimir invariants.

By antisymmetry of the bracket, $\gamma =f$ solves equation (\[INFNULL\]). Hence $\gamma = k(f)$ is also a solution, for an arbitrary function $k.$ Since the variation $k(f) \delta f$ is an exact variation (the formal analogue in infinite dimensions of an exact differential), the Pfaffian system, $k(f) \delta f = 0,$ is easily integrated. This yields the expression for the Casimir invariant ${\cal C}$:

$$\label{INFCAS.1}
{\cal C}[f] = \int _D K(f) {\rm d}^n z \, ,$$

where $K(f)$ is the primitive with respect to $f$ of $k(f).$ (Note: we use the caligraphic font to distinguish Casimir invariants of infinite-dimensional brackets from those of finte dimensional brackets.)

As another, slightly more complicated example, we consider a bracket of the form (\[INFBRAC\]) with an inner bracket of Lie-Poisson type. We suggest that brackets of this form are applicable to the Hamiltonian formulations of nontraditional kinetic theories in analogy to their use in the Vlasov-Poisson system mentioned above. Notice that the inner bracket for the Vlasov-Poisson system is that appropriate to the phase 