Integrals in involution

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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


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

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 $(x,u,p)$:


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

These definitions can be extended to a system of equations

$$F_i(x,u,p)=0,\quad1\leq i\leq m.\tag{3}$$

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

$$[F_i,G]=0,\quad1\leq i\leq m,\tag{4}$$

with unknown function $G$.

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


[1] N.M. Gyunter, "Integrating first-order partial differential equations" , Leningrad-Moscow (1934) (In Russian)
[2] E. Kamke, "Differentialgleichungen: Lösungen und Lösungsmethoden" , 2. Partielle Differentialgleichungen erster Ordnung für die gesuchte Funktion , Akad. Verlagsgesell. (1944)


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

How to Cite This Entry:
Integrals in involution. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by A.P. Soldatov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article