Eigenvalue Problem for Lagrangian Systems. II

Abstract

The quadratic Lagrangian eigenvalue problem [ω²I − ωiA − H]ξ = 0 for H and iA, completely continuous Hermitian operators in a Hilbert space E, H ≥ 0, is investigated. The problem is reduced to an equivalent linear eigenvalue problem for a single completely continuous Hermitian operator in the Hilbert space E × E, and existence and convergence theorems for the eigenvectors and variational properties of the eigenvalues for the original quadratic problem are easily obtained from standard theorems. The general solution of the associated time‐dependent problem ξ + Aξ + Hξ = 0 is obtained under the further restriction that E be finite dimensional. Necessary and sufficient conditions for stability are given.