Pfaffian constraint

${\displaystyle \sum _{s=1}^n{A}_{rs}d{u}_s+A_{r}dt=0;r=1,\dots ,L}$

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 ${\displaystyle {\mathbb{R}}^{1+n}}$.

Types

A Pfaffian constraint is integrable iff it is 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 catastatic iff ${\displaystyle A_r=0}$. Otherwise, it is acatastatic.

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

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

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.

The problem of Pfaff is nontrivial, because it is possible for two individually non-holonomic constraints to together create a holonomic constraint system. For example, on ${\displaystyle {\mathbb{R}}^3}$, the 1-forms ${\displaystyle dy-zdx=0}$ and ${\displaystyle dy-(z+1)dx=0}$ are both contact forms, thus maximally non-integrable, but a system containing both of them is integrable. Its integral manifolds are precisely the lines parallel to the z-axis.

Holonomic constraint

Given a holonomic system described by a set of holonomic constraint equations

${\displaystyle f_r(u_1,u_2,u_3,\dots ,u_n,t)=0;r=1,\dots ,L}$

where ${\displaystyle \{u_1,u_2,u_3,\dots ,u_n\}}$ are the n generalized coordinates that describe the system, and where ${\displaystyle L}$ is the number of equations in a system of constraints, we can differentiate by the chain rule for each equation:

${\displaystyle \sum _{s=1}^n\frac{\mathrm{\partial }{f}_r}{\mathrm{\partial }{u}_s}d{u}_s+\frac{\mathrm{\partial }{f}_r}{\mathrm{\partial }t}dt=0;r=1,\dots ,L}$

A Pfaffian constraint can be plotted as a field of surfaces of size

Nonholonomic constraint

If the constraint is nonholonomic, but first-order in the velocity, then it can also be written in Pfaffian form. That is, if in each$${\displaystyle f_r(u_1,u_2,u_3,\dots ,u_n,{\dot{u}}_1,\dots ,{\dot{u}}_n,t)=0;r=1,\dots ,L}$$the terms involving ${\displaystyle \dot{u}}$ are linear in them, then it is a Pfaffian constraint. For example, ${\displaystyle {\dot{u}}_1\cos u_3-{\dot{u}}_2\sin u_3-1=0}$ is equivalent to ${\displaystyle d{u}_1\cos u_3-d{u}_2\sin u_3-dt=0}$.

Non-Pfaffian constraint

If the constraint is not first-order in velocity, then it is not a Pfaffian constraint in general. For example, the constant-speed constraint ${\dot{u}}_1^2+{\dot{u}}_2^2-1=0$ is not Pfaffian. Similarly, ${\displaystyle {\dot{u}}_1{\dot{u}}_2-1=0}$ is not Pfaffian.

If the constraint is an inequality, then it is not Pfaffian. For example, the constraint for a pendulum with a soft string of length ${\displaystyle L}$ is ${\displaystyle x^2+y^2\le L^2}$.

If the constraint is not smooth, then it is not Pfaffian.

If the constraint is not restricted to position and velocity, then it is not Pfaffian in general. However, it might turn out to be equivalent to one. For example, consider a particle moving on the plane. If it is constrained to have constant curvature then ${\displaystyle \ddot{y}\dot{x}-\ddot{x}\dot{y}=\kappa {\left({\dot{x}}^2+{\dot{y}}^2\right)}^{3/2}}$, and it involves acceleration. However if we require the particle to have smooth and nonzero velocity, then it in fact moves on a circle, and thus it is a holonomic constraint.