Local Diffeomorphisms in Symplectic Space and Hamiltonian Systems with Constraints

TL;DR

This paper develops exact normal forms for local symplectic diffeomorphisms and applies them to classify initial singularities in constrained Hamiltonian systems, addressing a problem posed by Melrose.

Contribution

It introduces new normal forms for symplectic diffeomorphisms and provides a classification of singularities in constrained Hamiltonian systems.

Findings

Normal forms for local symplectomorphisms derived

Classification of first singularities in constrained Hamiltonian systems achieved

Addresses Melrose's problem on glancing hypersurfaces

Abstract

In this paper we obtain exact normal forms with functional invariants for local diffeomorphisms, under the action of the symplectomorphism group in the source space. Using these normal forms we obtain exact classification results for the first occurring singularities of Hamiltonian systems with one-sided constraints, a problem posed by R. B. Melrose in his studies of glancing hypersurfaces.