# Linear Hamiltonian Systems: Quadratic Integrals, Singular Subspaces and Stability

*2018, Volume 23, Number 1, pp. 26-46*

Author(s):

**Kozlov V. V.**

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.

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