Geodesic
Quadratic conservation laws for equations of mathematical physics
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 75 (2020) no. 3, pp. 445-494

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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.
Keywords: linear systems, Hilbert space, Hamiltonian system, quadratic invariants, equations of mathematical physics.
Mots-clés : Poisson bracket 
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V. V. Kozlov. Quadratic conservation laws for equations of mathematical physics. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 75 (2020) no. 3, pp. 445-494. http://geodesic.mathdoc.fr/item/RM_2020_75_3_a2/