Cartan's Topological Structure I

Many physical theories start with the assumption that a physical system can be described by a 1-form of Action, A, such as A= pdq - Hdt. Following Cartan's methods other p-forms can be constructed by forming the exterior derivative, dA, and all possible exterior products, A, dA, A^dA, dA^dA, ...

Pfaff Classes
Topological ActionA
Topological VorticitydA
Topological TorsionA^dA
Topological ParitydA^dA

The largest non-zero element of this Pfaff sequence determines the class, or Pfaff dimension, of the given exterior form. Remarkably, for 4 dimensions the set of elements, A, AUdA, A^dA,dAUdA,... forms a topological basis. The truthtable for this topology, defined as Cartan's topology, is given in the details

If the topological torsion vanishes, and the Pfaff dimension is less than three (A^dA=0), it turns out that the Cartan Topology is connected. However, if the topological torsion does not vanish, then the Cartan topology is a disconnected topology.

Recall that a topology is a method of determining a topological structure, whose utility is in giving a definition to what is meant by continuous. As it is impossible to construct a continuous map from a connected to a disconnected topology, it follows that a transition from a laminar flow (A^dA=0) to a turbulent flow (A^dA>0) must be a discontinuous process. On the other hand, a continous map from a disconnected topology to a connected topology is possible, therefor the decay of turbulence can be studied in terms of continuous processes.



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Last update 01/23/2009
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