Pfaffian systems

Definition

Following @bryant2013exterior page 43, it is an exterior differential system $\mathcal{E}$ algebraic-differentially generated by 1-forms $\{{\alpha }_1,\dots ,{\alpha }_r\}$. It can be identified with the finitely generated ${\mathcal{C}}^{\mathrm{\infty }}(M)$-submodule $\mathcal{P}=\mathcal{S}(\{{\alpha }_1,\dots ,{\alpha }_r\})$ of ${\mathrm{\Omega }}^1(M)$ (according to @conmorando page 4 or @PaolaPfaffian), since we can recover $\mathcal{E}$ by remixing ${\alpha }_i$ with $\wedge$ and $d$.

If the Pfaffian system is generated by only one 1-form $\omega$, it will be called a Pfaffian equation.

Completely integrable Pfaffian systems

The 2-forms

are important in order to understand the system. The Pfaffian system is called completely integrable when it is also algebraically generated (not only algebraic-differentially generated) by the 1-forms $\{{\alpha }_1,\dots ,{\alpha }_r\}$. In this case we have

From the "point of view of Paola", if we call Pfaffian system to the submodule $\mathcal{P}$, it will be completely integrable if the ideal algebraic-differentially generated by $\mathcal{P}$ (which is $\mathcal{E}$) is equal to the ideal algebraically generated by $\mathcal{P}$. Or, in the notation here, ${\mathcal{P}}_{alg}={\mathcal{P}}_{diff}$.

Another way of saying that it is completely integrable is requiring that the derived system

satisfies ${\mathcal{P}}^{\prime }=\mathcal{P}$.

At the beginning I thought (because of the approach of @warner) that a Pfaffian system was a finitely generated ${\mathcal{C}}^{\mathrm{\infty }}(M)$-submodule $\mathcal{P}$ of ${\mathrm{\Omega }}^1(M)$:

with ${\theta }^{\alpha }$ independent. And I thought that it was called completely integrable when the ideal $\mathcal{I}(\mathcal{P})$ of ${\mathrm{\Omega }}^{\ast }(M)$ is a differential ideal. At the end of the day, a completely integrable Pfaffian system is the same from both points of view.

For the visualization of this, see visualization of integrability of a Pfaffian system.

Integral manifolds

They have integral manifolds, according to Frobenius theorem dual version.

Dual distribution

A dual characterization of $\mathcal{P}$ is the subbundle $\mathcal{D}$ of $TM$ consisting of tangent vectors which are annihilated by ${\theta }^{\alpha }\in \mathcal{P}$, that is,

This is a distribution (see dual description of the distribution).

Linear Pfaffian system

Consider a Pfaffian system $\mathcal{E}=\{{\theta }^{\alpha }{\}}_{diff}$ with independence condition $\mathrm{\Omega }={\omega }^1\wedge \dots \wedge {\omega }^n$. We define $J:=\{{\theta }^{\alpha },{\omega }^i\}$. The Pfaffian system will be called linear if

and is usually denoted by $(I,J)$ where $I$ is the subbundle of $T^{\ast }M$ associated to $\mathcal{I}$.

The integral elements are determined by linear equations (I should work out an example, see example 3 in @landsberg1997exterior).

Two very important examples

Ordinary differential equations

It turns out that every ODE and, moreover, every system of ODEs, is a completely integrable Pfaffian system on a jet bundle. Indeed, they are rank 1 (hence involutive) distributions, which are trivially involutive. And involutivity is equivalent to complete integrability.
For example, the ODE $u_3=\varphi (x,u,u_1,u_2)$ is encoded in the exterior differential system generated by

and defined on $U\subseteq J^3(\mathbb{R},\mathbb{R})$. This is equivalent, by pulling back locally with $\iota :(x,u,u_1,u_2)\mapsto (x,u,u_1,u_2,\varphi )$, to the Pfaffian system

defined on $V\subseteq J^2(\mathbb{R},\mathbb{R})$.
Observe that

d(du_1-u_2 dx)=dx\wedge du_2=dx\wedge(du_2-\phi dx),

d(du_2-\phi dx)=dx\wedge (\phi_xdx + \phi_udu+\phi_{u_1}du_1+\phi_{u_2}du_2)=

=\phi_u dx\wedge(du-u_1 dx)+\phi_{u_1}dx\wedge(du_1-u_2 dx)+\phi_{u_2}dx\wedge(du_2-\phi dx).