Regul. Chaotic Dyn., 2018, Volume 23, Issue 1, Pages 26–46
Linear Hamiltonian Systems: Quadratic Integrals, Singular Subspaces and Stability
Valery V. Kozlov
Steklov Mathematical Institute, Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991 Russia
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.
Hamiltonian system, quadratic integrals, integral cones, degree of instability, quantum systems, Abelian integrals
|This work was carried out within the framework of the scientific program of the Presidium of the Russian Academy of Sciences 01 “Fundamental Mathematics and its Applications”.
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