Finding out normal coordinates with the method of undetermined coefficients: An alternative starting point of solving a small oscillation problem with two degrees of freedom

Abstract

A small oscillation with s degrees of freedom can be decomposed into s independent oscillations with one degree of freedom with the aid of normal coordinates. In the mainstream solution strategy, the normal coordinates are usually found out after the motion of the system has already been worked out using Lagrange’s equations. In this article, the search for the normal coordinates is used as the starting point of solving the problems with s = 2. For this purpose, the coefficients in the linear transformation from a set of generalized coordinates to a set of normal coordinates are treated as undetermined coefficients. The equations these undetermined coefficients must satisfy are deduced and solved. In comparison with the mainstream method, the dependence of the generalized coordinates on the time can be worked out in a more straightforward manner using the method of undetermined coefficients in the s = 2 case. Especially, even when the two angular frequencies degenerate, the two worked-out eigenvectors are still automatically orthogonal to each other with respect to the inertia matrix.