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]