Conical Singular Points and Vector Fields

We consider several examples of mechanical systems the configuration spaces of which have the form of smooth manifolds with a unique singular point: two intersecting (or tangent) curves on a two-dimensional torus, four curves on a four-dimensional torus with a common point, and a two-dimensional cone (cusp) in R⁶ . The main problem presented in this paper is the calculation of the (co)tangent space above the singular point by means of various theoretical approaches. Outside singular points, the motion of the mechanisms in question is described in the context of classical mechanics. However, in the neighborhood of a singular point, terms like “tangent vector” and “cotangent vector” must have conceptually new definitions. In this paper, the approach of the theory of “differential spaces” is used. In the case of a conical singular point, to calculate (co)tangent space, we use two various differential structures: the algebra of functions locally constant near the cone vertex and the algebra of restrictions of smooth functions from enveloping space to a cone. In the first case, tangent and cotangent spaces at the cone vertex are zero. In the second case, the algebra of functions on the cotangent bundle consists of functions locally constant on the cotangent layer above the singular point.