On contact Hamiltonian functions of singular contact structures

TL;DR

This paper investigates the properties of contact Hamiltonian functions in singular contact structures, establishing injectivity and providing explicit formulas for their inverses in specific singular cases.

Contribution

It extends the understanding of contact Hamiltonian functions to singular contact structures, offering explicit formulas where previously only the regular case was well-understood.

Findings

The map from infinitesimal contact transformations to Hamiltonian functions is injective in singular cases.

An explicit local formula for the inverse map is derived for singularities of the first type.

The study advances the theory of contact structures with singularities, especially regarding their Hamiltonian functions.

Abstract

For contact manifolds, it is well-known that the map which assigns to an infinitesimal contact transformation its contact Hamiltonian function is a linear isomorphism, and an explicit local formula for its inverse can be given. In contrast, infinitesimal contact transformations on a singular contact structure have not been well-studied yet. In this article, we show that this map is injective and present an explicit local formula for its inverse when the contact structure has singularities of the first type with structurally smooth Martinet hypersurface.