About the Weierstrass-Routh theorem

Abstract

As is known, Lagrange in his “Analytical Mechanics”, when presenting the theory of small oscillations of a system of points, made the incorrect statement that when the roots of the characteristic equation are multiplicity, the terms containing integer powers of time outside the signs of sine and cosine always appear in the solution. The error of this statement was indicated in 1858 by Weierstrass, who, however, considered only a special case of a system of equations containing terms with second-order derivatives and terms linear with respect to the required functions. Rouse, in his treatise on the stability of motion, considered the general case of equations also containing terms with first derivatives. This note is devoted to the same issue. Without containing significantly new results, it aims to provide an exhaustive study of the conditions under which the so-called secular terms are absent in the solution of a system of linear homogeneous equations of the 2nd order of the most general form. It is based on the Cauchy method of integrating linear equations, i.e. the same method that Weierstrass used when studying the above-mentioned special case. The advantage of this method is its naturalness: we look for expressions for the coefficients in the general integral of the system and, equating them to zero, obtain the required conditions.