There are several different ways to think about differential systems:
the general abstract way which we shall put forward here:
an exterior differential system is a sub-Lie-∞-algebroid $\mathfrak{a} \hookrightarrow T X$ of the tangent Lie algebroid $T X$ of a manifold that is the kernel of a morphism $p : T X \to \mathfrak{j}$ of Lie-∞-algebroids:
In the literature – the the references below – the term exterior differential system is instead introduced and understood in the context of dg-algebra as a dg-ideal $J \subset \Omega^\bullet(X)$ inside the deRham dg-algebra of $X$ and all concepts there are developed from this perspective.
From this the above perspective is obtained by noticing that from a dg-ideal $J$ we are naturally led to form the quotient dg-algebra $\Omega^\bullet(X)/J$ which is the cokernel of the inclusion $p^* : J \hookrightarrow \Omega^\bullet(X)$:
The existing literature on exterior differential systems is actually a bit unclear about which additional assumptions on $J$ are supposed to be a crucial part of the definition. However, in most applications of interest — see the examples below — it turns out that $J$ is in fact a semifree dga (over $C^\infty(X)$).
Here we take this as indication that
it makes good sense to understand exterior differential systems in the restricted sense where the dg-ideal $J$ is required to be a semifree dga;
the reason that the existing literature does present the desired extra assumptions on the dg-ideal $J$ in an incoherent fashion is due to a lack of global structural insight into the role of the definition of exterior differential systems.
Because, recall that a Lie-∞-algebroid is – effectively by definition – the formal dual of a semifree $\mathbb{N}$-graded commutative dg-algebra. So precisely with that extra condition on $J$ all dg-algebras in the above may be understood as Chevalley-Eilenberg algebras of Lie-∞-algebroids and then the above cokernel sequence of dg-algebras is precisely the formal dual of the kernel sequence of Lie-∞-algebroids.
Historically, one can trace back the basic idea of exterior differential systems to Eli Cartan’s work on partial differential equations in terms of differential forms:
for each system of partial differential equations
there is a space $X$ and a dg-ideal $J \in \Omega^\bullet(X)$ such that solutions of the system of equations are given by integral manifolds of the exterior differential system determined by $J$.
The notion of an integral manifold of an exterior differential system is crucial in the theory: in terms of $J$ it is a morphism $\phi : \Sigma \to X$ of manifolds such that the pullback of the ideal vanishes, $\phi^* J = 0$.
But this says precisely that $\phi$ extends to morphism of Lie-∞-algebroids
with $CE(\mathfrak{a}) = \Omega^\bullet(X)/J$ as above. Therefore the relevance of the notion of integral manifolds in the theory we take as another indication that exterior differential systems are usefully thought of as being about Lie-∞-algebroids.
An exterior differential system on a smooth manifold $X$ is a dg-ideal $J \subset \Omega^\bullet(X)$ of the deRham dg-algebra $\Omega^\bullet(X)$ of $X$.
Notice that $J$ being a dg-ideal means explicitly that
$\forall \theta\in J \subset \Omega^\bullet(X), \omega \in \Omega^\bullet(X): \theta \wedge \omega \in J$
the $\mathbb{N}$-grading $J = \oplus_{k \in \mathbb{N}} J_k$ on the dg-algebra $J$ is induced from that of $\Omega^\bullet(X)$ in that $J_k = J \cap \Omega^\bullet(k)$
$\forall \theta \in J \subset \Omega^\bullet(X) : d \theta \in J$
An integral manifold of an exterior differential system is a submanifold $\phi : Y \hookrightarrow X$ such that the restriction of all $\theta \in J$ to $Y$ vanishes: $\that|_Y = 0$.
In other words, for an integral manifold the pullback of the ideal $J$ along the inclusion map $\phi$ vanishes: $\phi^* J = 0$.
Often further assumptions are imposed on exterior differential systems. Here are some:
An exterior differential system is called finitely generated if there is a finite set $\{\theta_k \in \Omega^\bullet(X)\}$ of differential forms such that $J$ is the dg-ideal generated by these, so that
Often it is assumed that $J_0 = \mathbb{R}$.
Dually in terms of Lie-∞-algebroids this assumption means that $J$ is the Chevalley-Eilenberg algebra of a Lie-∞-algebroid that is just an L-∞-algebra.
A strict independence condition on an exterior differential system $J \subset \Omega^\bullet(X)$ is an $n$-form $\omega \in \Omega^n(X)$ for some $n$ such that
$\omega$ is decomposable into a wedge product of $n$ 1-forms mod $J^n$
$\omega$ is pointwise not an element of $J$.
For $(J, \omega)$ am exterior differential system with strict independence condition $\omega$, an integral manifold is now more restrictively an integral manifold $\phi : \Sigma \to X$ for $J$ but now such that $\phi^* \omega$ is a volume form on $\Sigma$ (i.e. pointwise non-vanishing).
Some special types of exterior differential systems carry their own names.
A Frobenius system is an exterior differential system $J \subset \Omega^\bullet(X)$ that is locally generated as a graded-commutative algebra from a set $\{\theta_j \in \Omega^1(U)\}_j$ of 1-forms.
Frobenius systems are in bijection with involutive subbundles of the tangent bundle of $X$, i.e. subbundles $E \hookrightarrow T X$ such that for $v,w \in \Gamma(E) \subset \Gamma(T X)$ also the Lie bracket of vector fields of $v$ and $w$ lands in $E$: $[v,w] \in \Gamma(E) \subset \Gamma(T X)$:
given a Frobenius system the sections of $\Gamma(E)$ are defined locally to be the joint kernel of the maps $\{\theta_i : \Gamma(T U) \to \mathbb{R}\}$.
given ab involutive subbundle $E$ the corresponding Frobenius system is the collection of 1-forms that vanishes on $E$:
$J = \{\theta \in \Omega^1(X) | \theta|_{E} = 0\}$.
Notice that the involutive subbundle may be thought of precisely as a sub-Lie algebroid
of the tangent Lie algebroid (i.e. as a sub Lie-∞-algebroid that happens to be an ordinary Lie algebroid). And indeed, the Chevalley-Eilenberg algebra of $E$ is the quotient $\Omega^\bullet(X)/J$ of the deRham dg-algebra by the Frobenius system:
A special case of a Lie algebroid corresponding to a Frobenius system is the vertical tangent Lie algebroid $T_{vert} Y$ of a map $\pi : Y \to X$. This corresponds to the subbundle $ker(\pi_*) \subset T Y$ of vertical vector fields on $Y$, with respecct to $\pi$. The corresponding Frobenius system is that of horizontal differential forms
and
is the dg-algebra of vertical differential forms with respect to $Y$.
This plays a central role in the theory of Ehresmann connections and Cartan-Ehresmann ∞-connection.
A system
of partial differential equations in terms of variables $\{x^\mu\}_{\mu = 1}^n$ and functions $\{f^i\}_{i = 1}^s$ of the form
is encoded by an exterior differential system on the 0-locus
of the $\{F^\rho\}_\rho$ (assuming that this is a manifold) with the dg-ideal $J = \langle \theta_i \rangle_i$ generated by the 1-forms
Namely a solution to the system of partial differential equations is precisely a section of the projection
which defined an integral manifold of the exterior differential system.
The standard textbook is
Course note are provided in