In the study of Lagrangian systems with constraints it becomes evident that the kinematic assumptions :

dx - Vdt = 0

and

dV - Adt = 0

are strong topological constraints on the domain of interest. Suppose that these differential 1-forms are not perfect, but have (perhaps small) fluctuations. The non-zero RHS may have physical significance.

So reconstruct a Lagrange Action principle and use Lagrange multipliers to include the constraints:

A = L(x,v,t)dt + alpha ( dx-Vdt) +beta (dV-Adt)

Such additions change the Pfaff dimension and the interpretation of the analysis dramatically.

In summary it would appear that there exist certain terms (like a pressure gradient) that are due to the anholonomic fluctuations in dx-Vdt.

Similarly there exist certain terms (like a temperature gradient) that are due to the anholonomic fluctuations in dV-Adt.

It is suggested that the source of the Casimir pressure is ultimately due to such anholonomic fluctuations in position, and the Davies-Unruh temperature is due to such anholonomic fluctuations in velocity.

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