  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.
