View source for Integrals in involution

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Solutions of differential equations whose [[Jacobi brackets|Jacobi brackets]] vanish identically. A function $G(x,u,p)$ of $2n+1$ variables $x=(x_1,\dots,x_n)$, $u$, $p=(p_1,\dots,p_n)$ is a first integral of the first-order partial differential equation

\begin{equation}F(x,u,p)=0,\label{1}\tag{1}\end{equation}

$$u=u(x_1,\dots,x_n),\quad p_i=\frac{\partial u}{\partial x_i},\quad1\leq i\leq n,$$

if it is constant along each [[Characteristic|characteristic]] of this equation. Two first integrals $G(x,u,p)$, $i=1,2$, are in involution if their Jacobi brackets vanish identically in $(x,u,p)$:

\begin{equation}[G_1,G_2]=0.\label{2}\tag{2}\end{equation}

More generally, two functions $G_1,G_2$ are in involution if condition \eqref{2} holds. Any first integral $G$ of equation \eqref{1} is in involution with $F$; the last function itself is a first integral.

These definitions can be extended to a system of equations

\begin{equation}F_i(x,u,p)=0,\quad1\leq i\leq m.\label{3}\tag{3}\end{equation}

Here the first integral of this system $G(x,u,p)$ can be regarded as a solution of the system of linear equations

\begin{equation}[F_i,G]=0,\quad1\leq i\leq m,\label{4}\tag{4}\end{equation}

with unknown function $G$.

If \eqref{3} is an [[involutional system]], then \eqref{4} is a [[complete system]]. It is in involution if the functions $F_i$ in \eqref{3} do not depend on $u$.

====Comments====
For additional references see [[Complete system|Complete system]]. An involutional system is usually called a system in involution.

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  N.M. Gyunter,   "Integrating first-order partial differential equations" , Leningrad-Moscow  (1934)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E. Kamke,   "Differentialgleichungen: Lösungen und Lösungsmethoden" , '''2. Partielle Differentialgleichungen erster Ordnung für die gesuchte Funktion''' , Akad. Verlagsgesell.  (1944)</TD></TR>
</table>

{{SourceattributionEOM|Integrals in involution}}
[[Category:Mathematics]]