Non-local variational mechanics—I stationarity conditions with one unknown

Abstract

(for Non-Local Variational Mechanics, I–VII). This series of seven papers extends the classical (local) variational mechanics so as to include those cases where the Lagrangian function (action density) involves integrals of the field variables and their derivatives as arguments. We thus obtain Euler equations that are non-local. By non-local we mean that the Euler equations involve the values of the fields and their derivatives at all points in the domain of the independent variables; the Euler equations become integro-differential equations rather than just differential equations. The non-local variational mechanics involving only one field variable is studied first. A reasonably straightforward algorithm is developed for the calculation of the Euler-Lagrange operator applied to Lagrangian function, the annihilation of which gives the non-local Euler equations. The properties of this operator are studied with particular emphases on characterizing those Lagrangian functions that are identically annihilated by the Euler-Lagrange operator. A detailed study is then given of those Lagrangian functions that result in linear Euler equations. This serves to point out the degree of generality afforded by the non-local Euler equations and to provide the basis for saying when a given linear equation can be realized as the Euler equation for some Lagrangian function. With one field variable this is possible only when the homogeneous part of the given equation defines a self-adjoint linear operator. Boundary conditions are considered next. It is shown that, for linear equations, a problem with non-natural boundary conditions can always be obtained from the Lagrangian function for the corresponding problem with natural boundary conditions by adding to the latter Lagrangian function a Lagrangian function that is identically annihilated by the Euler-Lagrange operator. This result is combined with earlier ones and the essential aspects of the non-local variational mechanics so as to obtain a general theorem concerning the representation of the eigenvalues of arbitrary nonlinear operators by means of a generalized Rayleigh quotient method. The non-local variational mechanics with several field variables is then developed; vectors of Euler-Lagrange operators, properties of the Euler-Lagrange operators, Lagrangian functions that are identically annihilated by the vector of Euler-Lagrange operators, etc. It is shown that the Euler-Lagrange operators give an automatic adjoint calculator, and the application of this method allows us to show that any linear system of integro-differential equations can be imbedded in a variational statement. This imbedding, in turn, leads to a simple method of obtaining necessary conditions for the existence of solutions to the Euler equations. An important aspect of this method is that it is the same for differential equations, integral equations, and integro-differential equations. A generalization of the notion of a momentum-energy complex (a matrix that contains the essential information concerning conservation laws) is obtained for the non-localizable fields and the associated strong and weak identities are derived. These identities are used to examine the question of quadratures of the Euler equations. Transformation properties of the Euler-Lagrange operators are considered where the field variables are subjected to transformations that depend on both the field variables and the independent variables. The treatment concludes with an extension of the Lagrange multiplier theorem to non-local variational problems with non-local constraints. This extension is such that it reduces to the classical results and provides a uniform method for a much wider class of problems than previously considered. For instance, if the problem is an isoparametric one, the Lagrange multiplier functions turn out to be equivalent to constants and the problem reduces to the usual Lagrange multiplier method, but without the prior assumption that the multipliers are constants. Whether or not the multipliers are constants is automatically dictated by the method and need not be assumed at the start.