On the variational methods of Ritz and Boussinesq

Abstract

In Ritz's method, as is known, the given differential equation is considered as the Euler equation arising from the variation of some integral, wherein the integrand of the latter represents a quadratic form in terms of the unknown function and its first derivative. It therefore seems natural to apply the "closure equation" to the question under consideration, and this will be the subject of the first section of the present article. However, before proceeding to this, it should be noted that the result thus obtained, which is identical to the result of the above-mentioned article, is not without interest, in our opinion, if only for the reason that in this way one obtains, albeit in a particular case, a proof of the convergence of Ritz's process even when the quadratic form under the integral sign to be varied is not a definite positive form.

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