The classic assumption made by almost all students of science is the idea that
the velocity field satisfies the equations, dx  Vdt = 0. From a topological
point of view, this idea of "kinematic perfection" imposes a very strong
topological constraint on the domain of interest.
Why not add to the 1form of Action the constraint as a possibly anholonomic
contribution with a lagrange multiplier, p.
A =Ldt => Ldt + p(dxVdt)
where dxVdt need not be zero.
Rearranging terms leads to the familar expression
A=pdx  (L  pV)dt => pdx  Hdt.
It is apparent that the Lagrange multiplier, p, plays the role of the momenta,
in the Cartan  Hilbert action formulation. If p is presumed to be canonical, hence deducible from the partial derivatives of L with respect to v, it follows that the Pfaff dimension of A is 2n+1. IF p is not assumed to be canonical, that the Pfaff dimension of A is 2n+2.
It takes an even dimensional space to produce a torsion current, such that
motion in the direction of the torsion current is thermodynamically
irreversible. In this sense, anholonomic fluctuations are the cause of
thermodynamic irreversibility.
