Conjugate Points Revisited and Neumann-Neumann Problems

The theory of conjugate points in the calculus of variations is reconsidered with a perspective emphasizing the connection to finite-dimensional optimization. The object of central importance is the spectrum of the second-variation operator, analogous to the eigenvalues of the Hessian matrix in finite dimensions. With a few basic properties of this spectrum, one can gain a new perspective on the classic result that “stability requires the lack of conjugate points”. Furthermore, we show how the spectral perspective allows the extension of the conjugate point approach to variants of the classic problems in the literature, such as problems with Neumann-Neumann boundary conditions.