Welcome to the new version of CaltechAUTHORS. Login is currently restricted to library staff. If you notice any issues, please email coda@library.caltech.edu
Published 1998 | Accepted Version
Journal Article Open

Poisson reduction for nonholonomic mechanical systems with symmetry

Abstract

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].

Additional Information

© 1998, The papers are being reviewed individually. Research partially supported by the DOE contract DE-FG0395-ER25251.

Attached Files

Accepted Version - KoMa1998.pdf

Files

KoMa1998.pdf
Files (259.8 kB)
Name Size Download all
md5:335769f59aa62354f79d3e0ee9fbfc10
259.8 kB Preview Download

Additional details

Created:
August 19, 2023
Modified:
October 20, 2023