Conical Singular Points and Vector Fields

We consider several examples of mechanical systems the configuration spaces which have form smooth manifolds with a unique singular point: two intersecting (or tangent) curves on two-dimensional torus, four four-dimensional torus common point, and cone (cusp) in $${{\mathbb{R}}^{6}}$$ . The main problem presented this paper is calculation (co)tangent space above point by means various theoretical approaches. Outside points, motion mechanisms question described context classical mechanics. However, neighborhood terms like “tangent vector” “cotangent must conceptually new definitions. In paper, approach theory “differential spaces” used. case conical to calculate space, we use differential structures: algebra functions locally constant near vertex restrictions from enveloping cone. first case, tangent cotangent at are zero. second bundle consists layer point.