Regions of possible motion in mechanical systems

Abstract: A method to study the topology of the integral manifolds basing on their projections to some other manifold of lower dimension is proposed. These projections are called the regions of possible motion and in mechanical systems arise in a natural way as the regions on a space of configuration variables. To classify such regions we introduce the notion of a generalized boundary of a region of possible motion and give the equation to find the generalized boundaries. The inertial motion of a gyrostat (the Euler--Zhukovsky case) is considered as an example. Explicit parametric equations of generalized boundaries are obtained. The investigation gives the main types of connected components of the regions of possible motion (including the sets of the admissible velocities over each point of the region). From this information, the phase topology of the case is established.