On a singular system of differential equations arising in variational methods

Abstract

In this paper an algebraic method for the reduction of a Singular System of Linear Differential Ordinary Equations (SODE) of order n, when the matrix A( t) of the coefficients of the highest derivatives of each variables is of rank k < n, is presented. By employing the Kantorovich method for the solution of a variational formulation of an elastic problem the author found a singular not normalizable SODE, that is a linear differential system of order n whose independent boundary conditions are less than n. The aim of the paper is to reduce algebraically the given SODE to an equivalent SODE of order less than n, without altering the structure of the original system too much (for example the symmetry). By determining the null space of the matrices A and A T the given system is reduced to a pair of problems: a differential system of order k and an independent algebraic system of order h = n − k. The following alternatives are possible: (1) the algebraic problem is incompatible and the solution of the given SODE does not exist; (2) the rank of the matrix of the algebraic system is equal to h (the algebraic solution is unique); (3) the algebraic solution is not unique. The discussion of the existence of the solution of the algebraic problem shows the conditions under which the given SODE can be reduced to a normalizable differential system of order k (i.e. depending only on k integration constants).