Manchester Geometry Seminar 2015/2016

In mechanics, constraints that restrict the possible configurations of the system are termed holonomic. A simple example is the fixed length of the rod of a pendulum. Mechanical systems with constraints on the velocities that do not arise as constraints on positions are called nonholonomic. These often arise in rolling systems, like a sphere rotating without slipping on a table.

The equations of motion for nonholonomic systems were shown to be described in terms of an almost Poisson bracket that fails to satisfy the Jacobi identity by Maschke and Van der Schaft in 1994. In this talk I will review the geometric construction of this bracket and show that in fact there is a whole family of brackets possessing the same properties. In the presence of symmetries, I will show that it is crucial to consider the different members in this family to determine if the reduced equations of motion allow a Hamiltonian formulation.