# Nonholonomic Mechanical Systems with Symmetry

### A. M. Bloch, P. S. Krishnaprasad, J. E. Marsden, R. M. Murray

*Archive for Rational Mechanics and Analysis*, 136(1):21-99

This work develops the geometry and dynamics of mechanical systems with nonholonomic
constraints and symmetry from the perspective of Lagrangian mechanics and with a view to
control theoretical applications. The basic methodology is that of geometric mechanics
applied to the formulation of Lagrange d'Alembert, generalizing the use of connections and
momentum maps associated with a given symmetry group to this case. We begin by formulating
the mechanics of nonholonomic systems using an Ehresmann connection to model the
constraints, and show how the curvature of this connection enters into Lagrange's
equations. Unlike the situation with standard configuration space constraints, the
presence of symmetries in the nonholonomic case may or may not lead to conservation laws.
However, the momentum map determined by the symmetry group still satisfies a useful
differential equation that decouples from the group variables. This momentum equation,
which plays an important role in control problems, involves parallel transport operators
and is computed explicitly in coordinates. An alternative description using a ``body
reference frame* relates part of the momentum equation to the components of the*
Euler-Poincar\'{e} equations along those symmetry directions consistent with the
constraints. One of the purposes of this paper is to derive this evolution equation for
the momentum and to distinguish geometrically and mechanically the cases where it is
conserved and those where it is not. An example of the former is a ball or vertical disk
rolling on a flat plane and an example of the latter is the snakeboard, a modified version
of the skateboard which uses momentum coupling for locomotion generation. We construct a
synthesis of the mechanical connection and the Ehresmann connection defining the
constraints, obtaining an important new object we call the nonholonomic connection. When
the nonholonomic connection is a principal connection for the given symmetry group, we
show how to perform Lagrangian reduction in the presence of nonholonomic constraints,
generalizing previous results which only held in special cases. Several detailed examples
are given to illustrate the theory.

- Preprint: http://www.cds.caltech.edu/~murray/preprints/cds94-013.pdf
- Project(s): Template:HTDB funding::Powell