Quadratic conservation laws for equations of mathematical physics
Валерий Васильевич Козлов
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.
Topics & Concepts
MathematicsConservation lawQuadratic equationHilbert spaceDegenerate energy levelsHamiltonian systemDifferential equationPartial differential equationSpace (punctuation)Independent equationSystem of linear equationsMathematical analysisApplied mathematicsPhysicsQuantum mechanicsComputer scienceGeometryOperating systemQuantum chaos and dynamical systemsNonlinear Waves and SolitonsNumerical methods for differential equations