## Linear Hamiltonian systems: quadratic integrals, singular subspaces and stability.(English)Zbl 1400.37061

Summary: 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.

### MSC:

 37J25 Stability problems for finite-dimensional Hamiltonian and Lagrangian systems 37J15 Symmetries, invariants, invariant manifolds, momentum maps, reduction (MSC2010) 37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
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