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 \ | + | 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 \ | + | 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.
Integrals in involution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Integrals_in_involution&oldid=44711