Complete integral

The solution , , , of a first-order partial differential equation

(1)

that depends on parameters and in the relevant region satisfies the condition

If is considered as an -parameter family of solutions, then the envelope of any -parameter subfamily distinguished by the condition , , is a solution to (1). Then the lines of contact between the surfaces given by the complete integral and the envelope are characteristics of (1). A complete integral can be used to describe the solution of the characteristic system of the ordinary differential equations corresponding to (1), and thus enables one to reverse Cauchy's method, which reduces the solution of (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

(2)

This system is a characteristic one for the Jacobi equation

(3)

If the complete integral for (3) is known, then the integrals of the canonical system (2) are given by the equations , , , where and 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)