Anholonomic Fluctuations

The classic training of an engineer or scientist often begins with the study of kinematics, where the concept of a Velocity vector in introduced as the limit of the derivative of a position vector R with respect to time. It is little appreciated that this primitive idea is a topological constraint on the domain of variables, {R,V,t} that is best written as dR-Vdt = 0. In this later format it is apparent that he space of variables is constrained by a Cartan exterior differential system consisting of N vanishing 1-forms.

On the otherhand if an initial space, R0, at parametric value t=t0, is mapped to a final space, R, at parametric value t=t, then given the map it is possible to deduce the relations

dR = [J]dR0 + Vdt.
The kinematic equations dR-Vdt = 0 follow if either the initial conditions are constant (dR0=0), or dR0 is an eigenvalue of the Jacobian matrix of the map with nulleigenvalue, or [J] is the zero matrix. The usual assumption is that the initial conditions are fixed.

In the more general situation, the fundamental equation is dR-Vdt = (an Error or flucutation 1-form) <> 0. The Error 1-forms can be interpreted as fluctuations which may have zero values on average, or are closed in a topological limit sense (which implies that each 1-form of constraint has a vanishing exterior derivative and is integrable). Other constraints are possible. The fluctuation 1-forms need not be integrable in the Frobenius sense. Hence the evolutionary process can not be represented by a single parameter group. Phenomenologically, dissipation is often attributed to such anholonomic fluctuations.