Frobenius integrability and the Pfaff Dimension

Suppose a physical system admits a 1-form of Action. That is, the functional form of the coefficients of the 1-form of Action are subsumed to be adequate to describe the physical system. (Note that many primitive unconstrained Lagrange systems start with a 1-form of Action, A = L(x,v,t)dt on a space N (=3n+3n+1) dimensions.

Now ask the question: Are all the independent variables necessary, or is there a minimal set of functions that can be used to describe the same problem.? Can functional combinations be constructed that yield the same topological information?

It is remarkable that there is a simple answer to such a question. The minimum number of functions required to adequately describe the 1-form is equal to its Pfaff dimension ( or class ).

Presume the orignal functional form of the 1-form has been constructed over some model space of variables, and presume the coefficient functions are c1 differentiable. Then construct the sequece of differential forms

{A, dA, A^dA, dA^dA, A^dA^dA,....}

At some integer M, the sequence terminates.

M is the Pfaff dimension of the 1-form.

Only M independent functions and there differentials are required to describe the system.

Topological Torsion appears at Pfaff dimension 3 and above. Thermodynamic irreversibility appears at Pfaff dimension 4.

If the Pfaff dimension is 2 or less, then A^dA =0, which is the Frobenious unique integrability condition. The direction field represented by the "normal" to the surface A=0 can be put into correspondence with the gradient of a "unique" function. Uniqueness in this sense is lost in domains where the Pfaff dimension is 3 or greater.

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