Event № 753
Nonholonomic mechanics concerns with mechanical systems whose velocity is constrained. If these velocity constraints are linear, they define k-planes at every point of the configuration space of the system. In more complex situations further constraints appear: the movement of the system not only has to be tangent to these k-planes, but must obey conditions in which tangent vectors to the trajectories have constant length, or satisfy other, in general nonlinear, relations. This equips kinematics of nonholonomic mechanical systems with various geometric structures. These are: vector distributions on manifolds, their symmetry groups, differential invariants, associated exterior differential systems, Cartan connections, etc.
In the lectures we will discuss these geometric structures in simple examples of existing (or possible to exist) mechanical systems. We will concentrate on systems whose kinematics is described by a low dimensional parabolic geometry i.e. a geometry modeled on a homogeneous space G/P, with G being a simple Lie group, and P being its parabolic subgroup. The considered systems will include a movement of ice skaters on an ice rink, rolling without slipping or twisting of rigid bodies, movements of snakes and ants, and even movements of flying saucers. Geometric relations between these exemplary nonholonomic systems will be revealed. An appearance of the simple exceptional Lie group G2 will be a repetitive geometric phenomenon in these examples.