Difference between revisions of "Complete integral"
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The solution $u(x,a)$, $x=(x_1,\dots,x_n)$, $a=(a_1,\dots,a_n)$, of a first-order partial differential equation | The solution $u(x,a)$, $x=(x_1,\dots,x_n)$, $a=(a_1,\dots,a_n)$, of a first-order partial differential equation | ||
| − | $$F\left(x_1,\dots,x_n,u,\frac{\partial u}{\partial x_1},\dots,\frac{\partial u}{\partial x_n}\right)=0,\tag{1}$$ | + | $$F\left(x_1,\dots,x_n,u,\frac{\partial u}{\partial x_1},\dots,\frac{\partial u}{\partial x_n}\right)=0,\label{1}\tag{1}$$ |
that depends on $n$ parameters $a_1,\dots,a_n$ and in the relevant region satisfies the condition | that depends on $n$ parameters $a_1,\dots,a_n$ and in the relevant region satisfies the condition | ||
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$$\det|u_{x_ia_k}|\neq0.$$ | $$\det|u_{x_ia_k}|\neq0.$$ | ||
| − | If $u(x,a)$ is considered as an $n$-parameter family of solutions, then the envelope of any $(n-1)$-parameter subfamily distinguished by the condition $a_i=\omega_i(t_1,\dots,t_{n-1})$, $1\leq i\leq n$, is a solution to \ | + | If $u(x,a)$ is considered as an $n$-parameter family of solutions, then the envelope of any $(n-1)$-parameter subfamily distinguished by the condition $a_i=\omega_i(t_1,\dots,t_{n-1})$, $1\leq i\leq n$, is a solution to \eqref{1}. Then the lines of contact between the surfaces given by the complete integral and the envelope are characteristics of \eqref{1}. A complete integral can be used to describe the solution of the characteristic system of the ordinary differential equations corresponding to \eqref{1}, and thus enables one to reverse Cauchy's method, which reduces the solution of \eqref{1} to that of the characteristic system. This approach is used in analytical mechanics, where one has to find the solution of a canonical system of ordinary differential equations |
| − | $$\frac{dx_i}{\partial t}=\frac{\partial H}{\partial p_i},\quad\frac{dp_i}{\partial t}=-\frac{\partial H}{\partial x_i},\quad1\leq i\leq n.\tag{2}$$ | + | $$\frac{dx_i}{\partial t}=\frac{\partial H}{\partial p_i},\quad\frac{dp_i}{\partial t}=-\frac{\partial H}{\partial x_i},\quad1\leq i\leq n.\label{2}\tag{2}$$ |
This system is a characteristic one for the Jacobi equation | This system is a characteristic one for the Jacobi equation | ||
| − | $$u_t+H\left(x_i,\dots,x_n,t,\frac{\partial u}{\partial x_1},\dots,\frac{\partial u}{\partial x_n}\right)=0.\tag{3}$$ | + | $$u_t+H\left(x_i,\dots,x_n,t,\frac{\partial u}{\partial x_1},\dots,\frac{\partial u}{\partial x_n}\right)=0.\label{3}\tag{3}$$ |
| − | If the complete integral $u=u(x_1,\dots,x_n,t,a_1,\dots,a_n)=a_0$ for \ | + | If the complete integral $u=u(x_1,\dots,x_n,t,a_1,\dots,a_n)=a_0$ for \eqref{3} is known, then the $2n$ integrals of the canonical system \eqref{2} are given by the equations $u_{a_i}=b_i$, $u_{x_i}=p_i$, $1\leq i\leq n$, where $a_i$ and $b_i$ are arbitrary constants. |
Revision as of 15:47, 14 February 2020
The solution $u(x,a)$, $x=(x_1,\dots,x_n)$, $a=(a_1,\dots,a_n)$, of a first-order partial differential equation
$$F\left(x_1,\dots,x_n,u,\frac{\partial u}{\partial x_1},\dots,\frac{\partial u}{\partial x_n}\right)=0,\label{1}\tag{1}$$
that depends on $n$ parameters $a_1,\dots,a_n$ and in the relevant region satisfies the condition
$$\det|u_{x_ia_k}|\neq0.$$
If $u(x,a)$ is considered as an $n$-parameter family of solutions, then the envelope of any $(n-1)$-parameter subfamily distinguished by the condition $a_i=\omega_i(t_1,\dots,t_{n-1})$, $1\leq i\leq n$, is a solution to \eqref{1}. Then the lines of contact between the surfaces given by the complete integral and the envelope are characteristics of \eqref{1}. A complete integral can be used to describe the solution of the characteristic system of the ordinary differential equations corresponding to \eqref{1}, and thus enables one to reverse Cauchy's method, which reduces the solution of \eqref{1} to that of the characteristic system. This approach is used in analytical mechanics, where one has to find the solution of a canonical system of ordinary differential equations
$$\frac{dx_i}{\partial t}=\frac{\partial H}{\partial p_i},\quad\frac{dp_i}{\partial t}=-\frac{\partial H}{\partial x_i},\quad1\leq i\leq n.\label{2}\tag{2}$$
This system is a characteristic one for the Jacobi equation
$$u_t+H\left(x_i,\dots,x_n,t,\frac{\partial u}{\partial x_1},\dots,\frac{\partial u}{\partial x_n}\right)=0.\label{3}\tag{3}$$
If the complete integral $u=u(x_1,\dots,x_n,t,a_1,\dots,a_n)=a_0$ for \eqref{3} is known, then the $2n$ integrals of the canonical system \eqref{2} are given by the equations $u_{a_i}=b_i$, $u_{x_i}=p_i$, $1\leq i\leq n$, where $a_i$ and $b_i$ are arbitrary constants.
Comments
The Jacobi equation is usually called the (time-dependent) Hamilton–Jacobi equation (see also Hamiltonian system).
References
| [a1] | P.R. Garabedian, "Partial differential equations" , Wiley (1964) |
| [a2] | B.A. Dubrovin, A.T. Fomenko, S.P. Novikov, "Modern geometry - methods and applications" , 1 , Springer (1984) (Translated from Russian) |
Complete integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Complete_integral&oldid=44720