This paper gives a general formulation of the theory of nonholonomic c
ontrol systems on a Riemannian manifold modeled by second-order differ
ential equations and using the unique Riemannian connection defined by
the metric. The main concern is to introduce a reduction scheme, repl
acing some of the second-order equations by first-order equations. The
authors show how constants of motion together with the nonholonomic c
onstraints may be combined to yield such a reduction. The theory is ap
plied to a particular class of nonholonomic control systems that may b
e thought of as modeling a generalized rolling ball. This class reduce
s to the classical example of a ball rolling without slipping on a hor
izontal plane.