Sections of Hamiltonian Systems

TL;DR

This paper classifies the typical singularities of sections in regular Hamiltonian systems, providing normal forms and relating them to symplectic classifications, advancing understanding of constrained Hamiltonian dynamics.

Contribution

It offers a complete list of normal forms for singularities of Hamiltonian system sections, linking them to symplectic classification of mappings with Whitney-type singularities.

Findings

Complete classification of singularities with normal forms

Connection between Hamiltonian sections and symplectic mappings

Insight into constrained Hamiltonian system behavior

Abstract

A section of a Hamiltonian system is a hypersurface in the phase space of the system, usually representing a set of one-sided constraints (e.g. a boundary, an obstacle or a set of admissible states). In this paper we give local classification results for all typical singularities of sections of regular (non-singular) Hamiltonian systems, a problem equivalent to the classification of typical singularities of Hamiltonian systems with one-sided constraints. In particular we give a complete list of exact normal forms with functional invariants, and we show how these are related/obtained by the symplectic classification of mappings with prescribed (Whitney-type) singularities, naturally defined on the reduced phase space of the Hamiltonian system.