The role of integrability in a large class of physical systems

TL;DR

This paper explores how the integrability of 1-forms influences the behavior of various physical systems, emphasizing the importance of normal forms and the Pfaff problem in understanding constraints and conservation laws.

Contribution

It provides a comprehensive mathematical discussion of the Pfaff problem and applies these concepts to multiple physical models, highlighting the role of integrability in energy conservation.

Findings

Integrability of 1-forms is crucial for understanding physical constraints.

Normal forms of 1-forms facilitate analysis of non-conservative forces and constraints.

Integrability impacts energy conservation across different physical systems.

Abstract

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.