Complete integral
The solution , 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) Zbl 0124.30501 |
[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=55538