In dynamics, a Pfaffian constraint is a way to describe a dynamical system in the form:

where ${\displaystyle L}$ is the number of equations in a system of constraints, and ${\displaystyle A_{rs},A_{r}}$ are functions of ${\displaystyle t,u_{1},\dots ,u_{n}}$ only. In other words, it is a 1-form on ${\displaystyle \mathbb {R} ^{1+n}}$.

A Pfaffian constraint is integrable iff it is a holonomic. Otherwise it is non-integrable or nonholonomic.

A Pfaffian constraint is scleronomous or scleronomic iff the coefficients ${\displaystyle A_{rs},A_{r}}$ do not depend on time. Otherwise it is rheonomous or rheonomic.

A Pffafian constraint is acatastatic iff ${\displaystyle A_{r}=0}$. Otherwise it is catastatic.

These together produce 8 types of Pffafian constraints, all of which are possible: {non-integrable, integrable} × {scleronomous, rheonomous} × {acatastatic, catastatic}.

A Pffafian constraint system is scleronomous/acatastic iff all its constraints are scleronomous/acatastic.

Pfaffian constraints are named after Johann Friedrich Pfaff, who studied the problem of Pfaff: Find the necessary and sufficient conditions for a Pfaffian constraint system to be integrable. In 1815, Pfaff published a general way to integrate first-order partial differential equations (PDE). The idea was to convert such a PDE in ${\displaystyle n}$ variables to a 1-form ${\displaystyle \omega }$ in ${\displaystyle \lceil n/2\rceil }$ variables, then integrate ${\displaystyle \omega }$. This problem was important in the development of modern differential geometry. Milestones include (Clebsch, 1866), (Frobenius, 1877), (Darboux, 1882), (Cartan, 1899). See integrability conditions for differential systems for the solution.