A regularization approach to non-smooth symplectic geometry

Abstract

We introduce non-smooth symplectic forms on manifolds and describe corresponding Poisson structures on the algebra of Colombeau generalized functions. This is achieved by establishing an extension of the classical map of smooth functions to Hamiltonian vector fields to the setting of non-smooth geometry. For mildly singular symplectic forms, including the continuous non-differentiable case, we prove the existence of generalized Darboux coordinates in the sense of a local non-smooth pull-back to the canonical symplectic form on the cotangent bundle.