The stability domains of hamiltonian systems

Linear Hamiltonian systems with an arbitrary number of degrees of freedom, which depend smoothly on a vector of real parameters, are investigated. All possible singularities of the boundary of the stability domain of Hamiltonian systems of general position are determined and described for the case of two and three parameters. In the first approximation, the geometry of these singularities (the orientation in the parameter space, angles, etc.) is determined on the basis of the first derivative of the matrix of the system with respect to the parameters, as are the eigenvectors and generalized eigenvectors evaluated at the singular point. A detailed investigation is made of gyroscopic systems as a special case of Hamiltonian systems. As mechanical examples, an account is given of the problem of the stability of the oscillations of a tube through which a fluid is flowing, and of the stability of the motion of a two-body system. The tangent cones to the stability domains of these systems at singular points of the “cusp” and “dihedral angle” type, which arise on the boundaries of these domains, are found.