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Solutions of differential equations whose [[Jacobi brackets|Jacobi brackets]] vanish identically. A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051700/i0517001.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051700/i0517002.png" /> variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051700/i0517003.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051700/i0517004.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051700/i0517005.png" /> is a first integral of the first-order partial differential equation
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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051700/i0517006.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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$$F(x,u,p)=0,\label{1}\tag{1}$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051700/i0517007.png" /></td> </tr></table>
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$$u=u(x_1,\dots,x_n),\quad p_i=\frac{\partial u}{\partial x_i},\quad1\leq i\leq n,$$
  
if it is constant along each [[Characteristic|characteristic]] of this equation. Two first integrals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051700/i0517008.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051700/i0517009.png" />, are in involution if their Jacobi brackets vanish identically in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051700/i05170010.png" />:
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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 $(x,u,p)$:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051700/i05170011.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
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$$[G_1,G_2]=0.\label{2}\tag{2}$$
  
More generally, two functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051700/i05170012.png" /> are in involution if condition (2) holds. Any first integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051700/i05170013.png" /> of equation (1) is in involution with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051700/i05170014.png" />; the last function itself is a first integral.
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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051700/i05170015.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
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$$F_i(x,u,p)=0,\quad1\leq i\leq m.\label{3}\tag{3}$$
  
Here the first integral of this system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051700/i05170016.png" /> can be regarded as a solution of the system of linear equations
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Here the first integral of this system $G(x,u,p)$ can be regarded as a solution of the system of linear equations
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051700/i05170017.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>
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$$[F_i,G]=0,\quad1\leq i\leq m,\label{4}\tag{4}$$
  
with unknown function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051700/i05170018.png" />.
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with unknown function $G$.
  
If (3) is an [[Involutional system|involutional system]], then (4) is a [[Complete system|complete system]]. It is in involution if the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051700/i05170019.png" /> in (3) do not depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051700/i05170020.png" />.
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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=15040
This article was adapted from an original article by A.P. Soldatov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article