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Difference between revisions of "Integrals in involution"

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Solutions of differential equations whose [[Jacobi brackets|Jacobi brackets]] vanish identically. A function 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
 
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
  
F(x,u,p)=0,\tag{1}
+
$$F(x,u,p)=0,\label{1}\tag{1}$$
  
 
u=u(x_1,\dots,x_n),\quad p_i=\frac{\partial u}{\partial x_i},\quad1\leq i\leq n,
 
u=u(x_1,\dots,x_n),\quad p_i=\frac{\partial u}{\partial x_i},\quad1\leq i\leq n,
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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):
 
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):
  
[G_1,G_2]=0.\tag{2}
+
$$[G_1,G_2]=0.\label{2}\tag{2}$$
  
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.
+
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
 
These definitions can be extended to a system of equations
  
F_i(x,u,p)=0,\quad1\leq i\leq m.\tag{3}
+
$$F_i(x,u,p)=0,\quad1\leq i\leq m.\label{3}\tag{3}$$
  
 
Here the first integral of this system G(x,u,p) can be regarded as a solution of the system of linear equations
 
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}
+
$$[F_i,G]=0,\quad1\leq i\leq m,\label{4}\tag{4}$$
  
 
with unknown function G.
 
with unknown function G.
  
If \ref{3} is an [[Involutional system|involutional system]], then \ref{4} is a [[Complete system|complete system]]. It is in involution if the functions F_i in \ref{3} do not depend on u.
+
If \eqref{3} is an [[Involutional system|involutional system]], then \eqref{4} is a [[Complete system|complete system]]. It is in involution if the functions F_i in \eqref{3} do not depend on u.
  
 
====References====
 
====References====

Revision as of 15:38, 14 February 2020

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

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

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):

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

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

F_i(x,u,p)=0,\quad1\leq i\leq m.\label{3}\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,\label{4}\tag{4}

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.

References

[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)


Comments

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: http://encyclopediaofmath.org/index.php?title=Integrals_in_involution&oldid=44711
This article was adapted from an original article by A.P. Soldatov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article