When physical systems admit description in terms of exterior differential forms, and when physical processes admit description in terms of singly parameterized vector fields, then it becomes apparent that whether or not the domain of indepedent variables is of even or odd dimension is crucial to the resulting solution possibilities. Given a 1form of Action describing a physical system, the Pfaff dimension is the irreducible number of functions required to describe the system over the domain. The original discription may be in the form of a large number (mole numbers) of variables, but the Pfaff dimension may be only 2n+1 or 2n+2 where n is a small integer. If the Pfaff dimension is odd the manifold upon which it is built will support a contact structure. If the Pfaff dimension is even, the manifold will support a symplectic structre.
On the odd dimensional contact structure (of maxial rank) there exists a unique
direction field completely determined by the components of the system. This
direction field forms the unique eigen vector with null eigen value for the
antisymmetric matrix of coefficients that define the 2form F=dA. This
extremal direction forms the "extremal" vector of the calculus of variations,
and evolution in the direction of this extremal field is conservative and admits
a Hamiltonian representation. When the 1form of virtual work W is defined as
W=i(v)F, it is apparent that the virtual work is zero for extremal Hamiltonian
evolution. Such is the case for the "force free" plasma.
When the Pfaff dimension of a physical system implies an even dimensional
symplectic structure, there do not exist any null eigen vectors of F=dA. Hence
the unique extremal evolutionary field does not exist on maximal rank domains of
even dimension. There still exist "Hamiltonian" flows, but these processes are
generated from a Bernoulli function, such that the Work 1form is a perfect
gradient. Such is the case for Eulerian hydrodynamics.
For the even dimensional case, there does exist a direction field that is
uniquely determined from the functions that define the physical system, A.
That vector field is defined as the "Torsion vector". Evolution of the Action A
in the direction of the torsion field is thermodynamically irreversible!!! In
fact, evolution of the Action in the direction of the Torsion vector is
proportional to itself. The function of proportionality is not constant and can
be attractive to or repelling from a zero set.
The sympectic (even dimensional) system decays by an irreversible (not
symplectic) process in the direction of the Torsion vector, A^F with a decay
rate determined by F^F. Once the system has decayed to the value 0, The Pfaff
dimension of the domain is odd, and subsequent evolution can be made by a
uniquely defined Hamiltonian process.
