Cartan developed the idea, given a 1form of Action describing a phsycial
system, that if the Pfaff dimension was 2n+1 (state space) then there existed a
unique vector field that had a null eigen value relative to the matrix of two
forms, F=dA. Such processes defined by this unique vector field are said to be
extremals (the variation of the action integral is zero along such directions.)
The extremal vector fields have a Hamiltonian representation, for which the
closed integral of Action (flux) is an evolutionary invariant and the open
integrals of F are evolutionary invariants. The latter concept corresponds to
Helmholtz conservation of vorticity theorem.
It the Pfaff dimension of the action 1form is even, the extremal field does not
exist (assuming maximal rank). However, there still can exist a Hamiltonian
representation (only now the Hamiltonian generating function is called the
CasimirBernoulli invariant) for the flow if the Virtual Work 1form, W=i(V)dA =
dT, is a perfect differential. An even more general situation that still
preserves the open integrals of F=dA as an evolutionary invariant is given by
those situations where the Work 1form is closed, dW=0. Such cases define a
symplectic evolutionary process. They are thermodynamically reversible, for Q^dQ
= L(V)A^L(V)dA= 0.
