Conjugate Points Revisited and Neumann–Neumann Problems

Abstract

Abstract. 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.

On Stability Analyses of Three Classical Buckling Problems for the Elastic Strut

“…Consequently, the sufficient condition recovers the classical result that a strut with fixed-fixed boundary conditions is stable provided β < π 2 . These results agree with those obtained using the condition J1 by Born [3], among others (see, e.g., [16]). …”

“…However, for the free-free strut (NN), a naive application of the Jacobi condition yields estimates which do not agree with classical buckling results. Manning [16] recently presented a modified version of J1 which accommodates the free-free boundary conditions by explicitly eliminating solutions of the type (21). However, it is not obvious a priori why J1, which works so successfully for the fixed-free and fixed-fixed cases, should fail for the free-free case.…”

“…1(a, b), the search for conjugate points produces results which are in agreement with the classical predictions of the critical load P crit . However, as noted recently by Manning [16], a similar analysis for the free-free case will lead to erroneous conclusions. 2 Surprised by this result, we were lead to examine an earlier necessary condition for the positive definiteness of the second variation that was developed by Legendre in 1786.…”

“…5, the Riccati equation ( (17) or (59) 1 ) is examined for all three classical buckling problems, and sufficient conditions for the stability of the straight and buckled configurations to the strut are established. Furthermore, the absence of a solution to the Riccati equation for the free-free strut is a clear indication of the degenerate nature of this case and the need for a modified treatment of the type proposed recently by Manning [16] (see also Kuznetsov and Levyakov [10,13]). We close the paper with a discussion of future applications to Legendre's treatment to the stability of branched equilibria of elastic rods such as those discussed in [5,19].…”

“…Dating to Born's seminal thesis [3] in 1906 and continuing to this day (cf. [8,12,15,16,23]), the main avenue to quantify whether or not the total energy is minimized is to use Jacobi's necessary condition. 1 That is, one examines an accessory variational problem and determines conditions for the nonexistence of conjugate points.…”