Quadratic conservation laws for equations of mathematical physics

Abstract

Abstract Linear systems of differential equations in a Hilbert space are considered that admit a positive-definite quadratic form as a first integral. The following three closely related questions are the focus of interest in this paper: the existence of other quadratic integrals, the Hamiltonian property of a linear system, and the complete integrability of such a system. For non-degenerate linear systems in a finite-dimensional space essentially exhaustive answers to all these questions are known. Results of a general nature are applied to linear evolution equations of mathematical physics: the wave equation, the Liouville equation, and the Maxwell and Schrödinger equations. Bibliography: 60 titles.