2012/10/14 by D. H. Delphenich, Delphenich, David
Engineering · Physics and Astronomy · #Advanced Differential Geometry Research #Advanced Thermodynamics and Statistical Mechanics #Control and Stability of Dynamical Systems #FOS: Physical sciences #Mathematical Physics (math-ph)
paper · pdf · doi:10.48550/arxiv.1210.4976
openalex publication_date 2012/10/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.