Errata: Restricted Quadratic Forms and Their Application to Bifurcation and Stability in Constrained Variational Principles
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The subjects of this investigation are the abstract properties and applications of restricted quadratic forms. The first part of the presentation resolves the following question: if L is a self-adjoint linear operator mapping a Hilbert space H into itself, and S is a subspace of H, when is the quadratic form $\langle {u,Lu} \rangle $ positive for any nonzero $u \in S$? In the second part of the presentation, restricted quadratic forms are further examined in the specific context of constrained variational principles; and the general theory is applied to obtain information on stability and bifurcation. Two examples are then solved: one is finite-dimensional and of an illustrative nature; the other is a longstanding problem in elasticity concerning the stability of a buckled rod. In addition to being a valuable analytical tool for isoperimetric problems in the calculus of variations, the tests described are amenable to numerical treatment.