It is suggested that the position of a rigid body is determined by means of ambiguous parameters (coordinates of a vector), whose dependence on position has no analytical properties (the dependence of a system of three unique parameters must have properties in the form of a coherent topological construction of the multiplicity of the position of a rigid body). This allows Euler's equations of dynamics to be joined by three kinematic relations, thus obtaining a closed system of six differential equations for determining the position of a rigid body as a function of time.