Friday, October 19, 2012

1210.4976 (David Delphenich)

The role of integrability in a large class of physical systems    [PDF]

David Delphenich
A large class of physical systems involves the vanishing of a 1-form on a manifold as a constraint on the acceptable states. This means that one is always dealing with the Pfaff problem in those cases. In particular, knowing the degree of integrability of the 1-form is often essential, or, what amounts to the same thing, its canonical (i.e., normal) form. This paper consists of two parts: In the first part, the Pfaff problem is presented and discussed in a largely mathematical way, and in the second part, the mathematical generalities thus introduced are applied to various physical models in which the normal form of a 1-form has already been implicitly introduced, such as non-conservative forces, linear non-holonomic constraints, the theory of vortices and equilibrium thermodynamics. The role of integrability in the conservation of energy is a recurring theme.
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