COMPLEMENTARY VARIATIONAL PRINCIPLES FOR BOUNDARY VALUE PROBLEMS I: BASIC PRINCIPLES, WITH AN APPLICATION TO ORDINARY DIFFERENTIAL EQUATIONS,

Abstract

In this series of papers a general variational principle and two complementary variational principles are developed for the simultaneous equations: Tp = partial derivative of W(x,p,u) with respect to u, and Su = partial derivative of W(x,p,u) with respect to p, where T,S are linear operators and S is the adjoint of T. The resulting formulae are of wide applicability and include most of the complementary variational principles which occur in applied mathematics. In this paper the implications of the variational principles are worked out in some detail when T = d/ dx, S = -d/ dx, including a discussion of extra terms which are required to deal with general boundary conditions.