Poisson structures on manifolds with singularities

TL;DR

This paper develops a method to analyze mechanical systems with singular configuration spaces by using algebraic structures derived from the geometry of singularities, enabling the application of Hamiltonian mechanics.

Contribution

It introduces an algebraic approach to define Poisson structures on manifolds with singularities, extending Hamiltonian mechanics to non-smooth configuration spaces.

Findings

Constructed Poisson structures on manifolds with one-dimensional singularities.

Applied the method to flat linkages and intersecting curves with contact points.

Enabled the use of differential operator theory on singular configuration spaces.

Abstract

Configuration spaces of many real mechanical systems appear to be manifolds with singularity. A singularity often indicates that geometry of motion may change at the singular point of configuration space. We face conceptual problem describing even mechanics of ideal models since the configuration space is not a smooth manifold, thus, the fully developed means of Hamiltonian Mechanics cannot be applied. In this report we present a way of conquering the aforementioned conceptual problem by considering a certain algebra instead of the configuration space. In this approach configuration space is the real spectrum of the algebra. The structure of this algebra is completely determined by the geometry of the singularity. For a broad class of singularities the desired algebra can be described directly since it is the pullback of two already known algebras. Availability of the algebra enables to use Differential Operator Theory. The elementary examples of mechanical systems to which this algorithm is applicable are flat linkages. In the frames of the presented approach we build a Poisson structure on a manifold with a one-dimensional singularity. The same result can be obtained to some other kinds of singularities. For instance,at the end the we give some specific results for the case of configuration space consisting of two curves on the plane with arbitrary order of contact at the intersection point.