129determining equations:6V 6p 6s 6gj6Xi6p= +P=VStX%j, p=mV, D,*p=O, D;(pXj) = O Dtaj = 0 , D,S=P*V-(H+h),(15)(16)3+=+ =+DJj =O,(17) (18)(19)+3(20)and, by 6A,, Maxwell equations with the currents (ep, epV). The vector field P = VSSXjVaj is now omplete�to represent any three-dimensional vectors if we take (at least) three independent scalars aj. The fluid (plasma) equation follows fromDtP = Dt(VS+ XjVaj) = -V(eq5 + h ) + 5 [V x B + (V . V)A] ,Cwhich is equivalent to (12). The role of S is best understood by referring to the original Serrin form ( X j = aj = 0 ) . Though Serrin S is a Lagrange multiplier that imposes mass conservation (ll),we proffer a different interpretation. By moving (by integrating by parts) D; from p to S,one may think of p as a Lagrange multiplier demanding that CF must be a complete derivative (evaluated through each streamline of V ) of some scalar field S -this is nothing but Hamilton principle demanding the criticality of the action integral with S as the ction)� Indeed, if the thermal energy E is neglected and X j = aj = 0 in (15) and (20), we obtain the well-known Hamilton-Jacobi equations atS = - H ( x , P , t ) and VS = P . The additional fields X j and a (further constraining the Lagrangian) j break the anonical structure�of the system, yielding eneralized Hamilton-Jacobi equations�(combining (15) and (20)):8,s = -vs = P - m a j .P , t ) + 14 - X3ataj,(21)(22)We have to specify what we have called on-canonical�. A canonical Hamiltonian mechanics is endowed with a regular (non-degenerate) Poisson bracket, i.e., if { F ,C} f 0 for all F , C must be a constant. Otherwise, i.e., if there exists such a non-constant C, which we call a Tasimir invariant� the system is non-canonical. The fluid-mechanics equations do have a Casimir invariant, that is the so-called elicity�defined by C = s ( V x P) . Pd2.596 Obviously, if P =
