Eigenvalue Problem for Lagrangian Systems. III

Abstract

The quadratic Lagrangian eigenvalue problem [λ²P + λQ ‐ (L + B)]ζ = 0 and the associated time‐dependent problem $\text{Pζ¨+iQζ̇+(L+B)ζ(t)=0}$ are investigated for the case where P, Q, and B are bounded linear Hermitian operators in Hilbert space, P is positive and invertible, L possesses a positive completely continuous Hermitian inverse, and L + B > 0. Existence and completeness theorems for the eigenvectors as well as variational characterizations of the eigenvalues are given, and the general solution of the time‐dependent problem is obtained in terms of an eigenvector expansion. Finally, these results are applied to the problem of small oscillations of a rotating elastic string.