This paper continues the work of Koon and Marsden [1997b] that began
the comparison of the Hamiltonian and Lagrangian formulations of
nonholonomic systems. Because of the necessary replacement of
conservation laws with the momentum equation, it is natural to let the
value of momentum be a variable and for this reason it is natural to
take a Poisson viewpoint. Some of this theory has been started in van
der Schaft and Maschke [1994]. We build on their work,
further develop the theory of nonholonomic Poisson reduction, and tie
this theory to other work in the area. We use this reduction
procedure to organize nonholonomic dynamics into a reconstruction
equation, a nonholonomic momentum equation and the reduced Lagrange
d'Alembert equations in Hamiltonian form. We also show that these
equations are equivalent to those given by the Lagrangian reduction
methods of Bloch, Krishnaprasad, Marsden and Murray [1996]. Because of
the results of Koon and Marsden [1997b], this is also equivalent to
the results of Bates and Sniatycki [1993], obtained by nonholonomic
symplectic reduction.
Two interesting complications make this effort especially
interesting. First of all, as we have mentioned, symmetry
need not lead to conservation laws but rather to a momentum
equation. Second, the natural Poisson bracket fails to
satisfy the Jacobi identity. In fact, the so-called
Jacobiizer (the cyclic sum that vanishes when the Jacobi
identity holds), or equivalently, the Schouten bracket, is an
interesting expression involving the curvature of the underlying
distribution describing the nonholonomic constraints.
The Poisson reduction results in this paper are important for the future
development of the stability theory for nonholonomic mechanical systems
with symmetry, as begun by Zenkov, Bloch and Marsden [1997]. In
particular, they should be useful for the development of the powerful
block diagonalization properties of the energy-momentum method
developed by Simo, Lewis and Marsden [1991].