Seminarium "The Trans-Carpathian Seminar on Geometry & Physics"

The configuration spaces of many real mechanical systems appear to bemanifolds with singularity. A singularity often indicates that the geometryof motion may change at the point. In such cases we face the conceptualproblem describing mechanics even for ideal models: since the configurationspace is not a smooth manifold, thus, the fully developed machinery ofHamiltonian mechanics cannot be applied.I will present a way of conquering the aforementioned conceptual problem byconsidering a certain algebra instead of the configuration space.Configuration space is the real spectrum of the algebra. The structure ofthis algebra is completely determined by the geometry of the singularity.For a broad class of singularities the desired algebra can be describeddirectly since it is the pullback of two already known algebras which canbe easily described. Availability of the algebra enables to useDifferential operator theory.The elementary examples of mechanical systems to which this algorithm isapplicable, are flat linkages.In the frames of the presented approach a Poisson structure is built on amanifold with a one-dimensional singularity. The same result can beobtained for some other kinds of singularities.At the end of the talk I will present the specific results for the case ofa configuration space consisting of two curves on the plane having randomorder of contact.