A geometric classification of Hamiltonian equilibria by spectral types

We give one algebraic equation and two semialgebraic equations whose corresponding semi-algebraic varieties are associated to definite types of bifurcations for parameter- dependent Hamiltonian systems. These equations provide a geometric classification of the spectral types of equilibria of a given m-degrees of freedom Hamiltonian system. In fact the complement of such varieties is the disconnected union of open domains (in parameter space) in which the spectral type of the equilibrium does not change. We analyse such varieties thoroughly, in the space of invariants, for generic systems with 2,3,4 degrees of freedom. This technique can be applied to any parameter-dependent Hamiltonian system, and it does give the bifurcation-decomposition of parameter space. We conclude with applications to two problems in classical mechanics.