Integrals in involution

Solutions of differential equations whose 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 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[edit]

For additional references see Complete system. An involutional system is usually called a system in involution.

#### References[edit]

[1] |  N.M. Gyunter, "Integrating first-order partial differential equations" , Leningrad-Moscow (1934) (In Russian)

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[2] |  E. Kamke, "Differentialgleichungen: Lösungen und Lösungsmethoden" , 2\. Partielle Differentialgleichungen erster Ordnung für die gesuchte Funktion , Akad. Verlagsgesell. (1944)