Reply by Author to C.V. Smith Jr. and D.R. Smith

Abstract

of the approximate solution if one blindly extended the integrations to r = 28 sec using the same values of tI and TV as used for smaller values of tl We believe that Ref. 4, when properly interpreted, is concerned with how one can achieve an approximate solution to motion problems when the differential equations can not be solved exactly. This has always been possible through the method of weighted residuals or the principle of virtual work. Indeed, Professor Bailey's applications of his Eq. (4) are simply Galerkin method approximate solutions of Lagrange's equations of motion after some integration by parts. Professor Bailey's presentation certainly serves to call attention to the need in his Eq. (4) for (dTVdtf, )6<?, when dq, * 0; but the content of Eq. (4) is not one whit different from the Galerkin solution. Furthermore, as mentioned in Ref. 3, Hamilton's Law of Varying Action as presented in 1834 and 1835 is quite different from what Professor Bailey describes with the same title in his Eq. (4). His procedure is not contrary to the state of energy theory found in textbooks and in the variational calculus, and he has not cleared up a 140 year mystery on how to achieve direct solutions. His contribution has been to provide some very excellent and informative examples of the application of Galerkin's method, demonstrating the accuracy possible for certain problems.