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 1-form of Action the constraint as a possibly anholonomic contribution with a lagrange multiplier, p.

A =Ldt => Ldt + p(dx-Vdt)

where dx-Vdt 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.

chaos97.pdf

As I have time I will replace this under construction symbol with a link to a more detailed pdf or tex file download.