Linear Hamiltonian Systems: Quadratic Integrals, Singular Subspaces and Stability
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A chain of quadratic first integrals of general linear Hamiltonian systems that have not been represented in canonical form is found. Their involutiveness is established and the problem of their functional independence is studied. The key role in the study of a Hamiltonian system is played by an integral cone which is obtained by setting known quadratic first integrals equal to zero. A singular invariant isotropic subspace is shown to pass through each point of the integral cone, and its dimension is found. The maximal dimension of such subspaces estimates from above the degree of instability of the Hamiltonian system. The stability of typical Hamiltonian systems is shown to be equivalent to the degeneracy of the cone to an equilibrium point. General results are applied to the investigation of linear mechanical systems with gyroscopic forces and finite-dimensional quantum systems.
KeywordsHamiltonian system quadratic integrals integral cones degree of instability quantum systems Abelian integrals
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- 13.Berger, M., Geometry: In 2 Vols., Berlin: Springer, 1987.Google Scholar
- 19.Takhtajan, L.A., Quantum Mechanics for Matematicians, Grad. Stud. Math., vol. 95, Providence, R.I.: AMS, 2008.Google Scholar
- 21.Kozlov, V.V., Topological Obstructions to Existence of the Quantum Conservation Laws, Dokl. Math., 2005, vol. 71, no. 2, pp. 300–302; see also: Dokl. Akad. Nauk, 2005, vol 401, no. 5, pp. 603–606.Google Scholar
- 22.Faddeev, L.D., What Is Complete Integrability in Quantum Mechanic, in Nonlinear Equations and Spectral Theory, M. S. Birman, N.N.Uraltseva (Eds.), Amer. Math. Soc. Transl. Ser. 2, vol. 220, Providence, R.I.: AMS, 2007, pp. 83–90.Google Scholar
- 25.Klein, F., Lectures on the Icosahedron and the Solution of Equations of the Fifth Degree, Moscow: Nauka, 1989 (Russian).Google Scholar