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{{HTDB paper
| authors = A. M. Bloch, P. S. Krishnaprasad, J. E. Marsden, R. M. Murray
| title = Nonholonomic Mechanical Systems and Symmetry
| source = CDS Technical Report 94-013<br>To appear <i>Archive for Rational Mechanics and Analysis</i>
| year = 1994
| type = CDS Technical Report
| funding =
| url = http://www.cds.caltech.edu/~murray/preprints/cds94-013.pdf
| abstract =
<h3>Abstract</h3>
This work develops the geometry and dynamics of mechanical systems
with nonholonomic constraints and symmetry from the point of view of
Lagrangian mechanics and with a view to control theoretical
applications. The basic methodology is that of geometric mechanics
emphasizing the formulation of Lagrange d'Alembert with the use of
connections and momentum maps associated with the given symmetry
group. We begin by recalling and extending the results of Koiller from
the case of principal connections to the general Ehresmann
case. Unlike the situation with standard configuration space
constraints, the symmetry in the nonholonomic case may or may not lead
to conservation laws. In any case, the momentum map determined by the
symmetry group satisfies a useful differential equation that decouples
from the group variables. This momentum equation is shown to have the
form of a covariant derivative of the momentum equal to a component of
the internal generalized force. An alternative description using a
"body reference frame" realizes part of the momentum equation as
those components of the Euler-Poincare equations along the
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, the nonholonomic connection. Under conditions that include the
Chaplygin case (we use the terminology "purely kinematic") and the
case in which the momentum is conserved, it is known that one can
perform a reduction similar to Lagrangian reduction, which includes
the Routh procedure. We generalize this reduction procedure to the
case in which the nonholonomic connection is a principal connection
for the given symmetry group; this case includes all of the examples
considered in the paper and many others as well, such as the
wobblestone, the nonvertical disk and the bicycle. Another purpose of
this work is to lay the foundation for future work on mechanical
systems with control so that one can adapt well developed techniques
from holonomic systems, such as constructive controllability and
geometric phases. Although this will be the subject of future work,
the methodology of the present paper is developed with these goals in
mind.
| flags = NoRequest
| tag = bkmm94-cds
| id = 1994g
}}

Nonholonomic Mechanical Systems and Symmetry - Revision history

Murray: htdb2wiki: creating page for 1994g_bkmm94-cds.html