# Involutional system

system in involution, involutive system of partial differential equations

A system of first-order partial differential equations

$$\tag{1 } F _ {i} ( x , u , p ) = 0 ,\ \ 1 \leq i \leq m ,$$

where $x = ( x _ {1} \dots x _ {n} )$, $u= u ( x _ {1} \dots x _ {n} )$, $p = ( p _ {1} \dots p _ {n} ) = ( \partial u / \partial x _ {1} \dots \partial u / \partial x _ {n} ),$ for which all Jacobi brackets are equal to zero:

$$\tag{2 } [ F _ {i} , F _ {j} ] = 0 ,\ \ 1 \leq i , j \leq m ,$$

identically in $( x , u , p )$. The equations (2) are called the integrability conditions.

This definition is somewhat modified for quasi-linear systems. Suppose that none of the functions $\partial F _ {i} / \partial p _ {k}$, $1 \leq i \leq m$, $1 \leq k \leq n$, depends on $p = ( p _ {1} \dots p _ {n} )$. Then the functions

$$[ F _ {i} , F _ {j} ] - F _ {i} \frac{\partial F _ {j} }{\partial u } + F _ {j} \frac{\partial F _ {i} }{\partial u }$$

also have this property. In the class of quasi-linear equations the condition for a system to be in involution (involutional) is defined by the equations

$$[ F _ {i} , F _ {j} ] - F _ {i} \frac{\partial F _ {j} }{\partial u } + F _ {j} \frac{\partial F _ {i} }{\partial u } = 0 ,\ \ 1 \leq i , j \leq m .$$

When the $F _ {i}$ do not depend on $u$, the definition is the same as the previous one. Sometimes the latter definition is extended to all systems of the form (1).

If the system (1) is linear and homogeneous and is expressed in the form

$$P _ {i} ( u) = 0 ,\ \ 1 \leq i \leq m ,$$

where the $P _ {i}$ are first-order linear differential operators, then being in involution can be defined for it as the commutativity condition $P _ {i} P _ {j} = P _ {j} P _ {i}$ for all $1 \leq i , j \leq m$.

Every system in involution is a complete system. Conversely, if (1) is a complete system and is in normal form, that is, $1 \leq m \leq n$ and

$$F _ {i} ( x , u , p ) = \ p _ {i} - f _ {i} ( x , u , p _ {m+1} \dots p _ {n} ) ,\ \ 1 \leq i \leq m ,$$

then it is in involution. This enables one to reduce a complete system to a system in involution if $m \leq n$, and to solve it by a non-singular transformation with respect to $m$ of the variables $p = ( p _ {1} \dots p _ {n} )$.

If the system (1) does not depend on $u$, if $m = n$, if the determinant $| \partial F _ {i} / \partial p _ {k} | \neq 0$, and if $p _ {i} = p _ {i} ( x)$ are solvable from the equations $F _ {i} ( x , p ) = 0$, $1 \leq i \leq n$, then if this system is in involution, the expression

$$\sum_{i=1} ^ { n } p _ {i} ( x) d x _ {i}$$

is an exact form. The application of Jacobi's method [2] for solving systems is involution not dependent on $u$ and consisting of $m$ functional unknown equations, $m < n$, is based on this. According to this method, the original system is extended to a system in involution of $n$ equations with the above properties. The extension proceeds in several stages: each successive system is obtained from the previous one by adding its unknown first integrals. This method also finds application for systems of equations depending on $u$( see [3]).

#### References

 [1] C. Carathéodory, "Calculus of variations and partial differential equations of the first order" , 1 , Holden-Day (1965) (Translated from German) [2] C.G.J. Jacobi, "Nova methodus, aequationes differentiales partiales primi ordinis inter numerum variabilium qeumcunque propositas integrandi" J. Reine Angew. Math. , 60 (1862) pp. 1–181 [3] E. Coursat, "Leçons sur l'intégration des équations aux dérivées partielles du premier ordre" , Hermann (1891) [4] N.M. Gyunter, "Integrating first-order partial differential equations" , Leningrad-Moscow (1934) (In Russian) [5] E. Kamke, "Differentialgleichungen: Lösungen und Lösungsmethoden" , 2. Partielle Differentialgleichungen erster Ordnung für die gesuchte Funktion , Akad. Verlagsgesell. (1944)

More generally, a system of partial differential equations is defined as follows. Let $\pi : M \rightarrow N$ be a fibre manifold, i.e. locally (up to diffeomorphisms) $\pi$ looks like a standard projection $\mathbf R ^ {n} \times \mathbf R ^ {m} \rightarrow \mathbf R ^ {n}$. Let $J ^ {l} = J ^ {l} ( \pi )$ be the $l$- th jet manifold of $\pi$ and let ${\mathcal O} ^ {l}$ be the sheaf of germs of functions on $J ^ {l}$. Then a system of partial differential equations of order $l$ on $U \subset J ^ {l}$ is a locally finitely-generated sheaf of ideals $\Phi$ of ${\mathcal O} ^ {l}$ restricted to $U$. A solution is a cross section $f : \pi ( U) \rightarrow U$ such that $\phi ( j _ {x} ^ {l} f ) = 0$ for all $\phi \in \Phi$, where $j _ {x} ^ {l} f$ is the $l$- jet of $f$ at $x$ in $\pi ( U)$.