In dynamics, a Pfaffian constraint is a way to describe a dynamical system in the form:
∑ s = 1 n A r s d u s + A r d t = 0 ; r = 1 , … , L {\displaystyle \sum _{s=1}^{n}A_{rs}du_{s}+A_{r}dt=0;\;r=1,\ldots ,L}
where L {\displaystyle L} is the number of equations in a system of constraints, and A r s , A r {\displaystyle A_{rs},A_{r}} are functions of t , u 1 , … , u n {\displaystyle t,u_{1},\dots ,u_{n}} only. In other words, it is a 1-form on R 1 + n {\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 A r s , A r {\displaystyle A_{rs},A_{r}} do not depend on time. Otherwise, it is rheonomous, or rheonomic. A Pffafian constraint is catastatic iff A r = 0 {\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 n {\displaystyle n} variables to a 1-form ω {\displaystyle \omega } in ⌈ n / 2 ⌉ {\displaystyle \lceil n/2\rceil } variables, then integrate ω {\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 R 3 {\displaystyle \mathbb {R} ^{3}} , the 1-forms d y − z d x = 0 {\displaystyle dy-zdx=0} and d y − ( z + 1 ) d x = 0 {\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
f r ( u 1 , u 2 , u 3 , … , u n , t ) = 0 ; r = 1 , … , L {\displaystyle f_{r}(u_{1},u_{2},u_{3},\ldots ,u_{n},t)=0;\;r=1,\ldots ,L}
where { u 1 , u 2 , u 3 , … , u n } {\displaystyle \{u_{1},u_{2},u_{3},\ldots ,u_{n}\}} are the n generalized coordinates that describe the system, and where L {\displaystyle L} is the number of equations in a system of constraints, we can differentiate by the chain rule for each equation:
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