  The nonholonomic dynamics can be described by the so-called nonholonomic
bracket on the constrained submanifold, which is a non-integrable modification
of the Poisson bracket of the ambient space, in this case, of the canonical
bracket on the cotangent bundle of the configuration manifold. On the other
hand, another bracket, also called nonholonomic bracket, was defined using the
description of the problem in terms of skew-symmetric algebroids. Recently,
reviewing two older papers by R. J. Eden, we have defined a new bracket which
we call Eden bracket. In the present paper, we prove that these three brackets
coincide. Moreover, the description of the nonholonomic bracket \`a la Eden has
allowed us to make important advances in the study of Hamilton-Jacobi theory
and the quantization of nonholonomic systems.
