Symplectic Singularities and Solvable Hamiltonian Mappings

Abstract

We study singularities of smooth mappings F̄ of ℝ2n into symplectic space (ℝ2n , ω̇) by their isotropic liftings to the corresponding symplectic tangent bundle (Tℝ2n,w). Using the notion of local solvability of lifting as a generalized Hamiltonian system, we introduce new symplectic invariants and explain their geometric meaning. We prove that a basic local algebra of singularity is a space of generating functions of solvable isotropic mappings over F̄ endowed with a natural Poisson structure. The global properties of this Poisson algebra of the singularity among the space of all generating functions of isotropic liftings are investigated. The solvability criterion of generalized Hamiltonian systems is a strong method for various geometric and algebraic investigations in a symplectic space. We illustrate this by explicit classification of solvable systems in codimension one