Research Papers

Reduction of Hamiltonian Mechanical Systems With Affine Constraints: A Geometric Unification

[+] Author and Article Information
Robin Chhabra

MacDonald, Dettwiler and Associates Ltd.,
9445 Airport Rd,
Brampton, Ontario L6S 4J3, Canada
e-mail: robin.chhabra@mdacorporation.com

M. Reza Emami

Professor and Chair
Department of Computer Science,
Electrical, and Space Engineering,
Luleå University of Technology,
Luleå 971 87, Sweden;
Institute for Aerospace Studies,
University of Toronto,
4925 Dufferin Street,
Toronto, Ontario M3H 5T6, Canada
e-mail: reza.emami@ltu.se

Yael Karshon

Department of Mathematics,
University of Toronto,
40 St. George Street 6th Floor,
Toronto, Ontario M5S 2E4, Canada
e-mail: karshon@math.toronto.edu

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received December 25, 2015; final manuscript received August 16, 2016; published online December 2, 2016. Assoc. Editor: Zdravko Terze.

J. Comput. Nonlinear Dynam 12(2), 021007 (Dec 02, 2016) (14 pages) Paper No: CND-15-1456; doi: 10.1115/1.4034729 History: Received December 25, 2015; Revised August 16, 2016

This paper presents a geometrical approach to the dynamical reduction of a class of constrained mechanical systems. The mechanical systems considered are with affine nonholonomic constraints plus a symmetry group. The dynamical equations are formulated in a Hamiltonian formalism using the Hamilton–d'Alembert equation, and constraint forces determine an affine distribution on the configuration manifold. The proposed reduction approach consists of three main steps: (1) restricting to the constrained submanifold of the phase space, (2) quotienting the constrained submanifold, and (3) identifying the quotient manifold with a cotangent bundle. Finally, as a case study, the dynamical reduction of a two-wheeled rover on a rotating disk is detailed. The symmetry group for this example is the relative configuration manifold of the rover with respect to the inertial space. The proposed approach in this paper unifies the existing reduction procedures for symmetric Hamiltonian systems with conserved momentum, and for Chaplygin systems, which are normally treated separately in the literature. Another characteristic of this approach is that although it tracks the structure of the equations in each reduction step, it does not insist on preserving the properties of the system. For example, the resulting dynamical equations may no longer correspond to a Hamiltonian system. As a result, the invariance condition of the Hamiltonian under a group action that lies at the heart of almost every reduction procedure is relaxed.

Copyright © 2017 by ASME
Your Session has timed out. Please sign back in to continue.


Marsden, J. E. , and Ratiu, T. S. , 1999, Introduction to Mechanics and Symmetry, Springer-Verlag, New York.
Bloch, A. M. , Krishnaprasad, P. S. , Marsden, J. E. , and Murray, R. M. , 1996, “ Nonholonomic Mechanical Systems With Symmetry,” Arch. Ration. Mech. Anal., 136(1), pp. 21–99. [CrossRef]
Bloch, A. M. , 2003, Nonholonomic Mechanics and Control, Springer, New York.
Koon, W. S. , and Marsden, J. E. , 1997, “ The Hamiltonian and Lagrangian Approaches to the Dynamics of Nonholonomic Systems,” Rep. Math. Phys., 40(1), pp. 21–62. [CrossRef]
Chhabra, R. , and Emami, M. R. , 2016, “ A Unified Approach to Input-Output Linearization and Concurrent Control of Underactuated Open-Chain Multi-Body Systems With Holonomic and Nonholonomic Constraints,” J. Dyn. Control Syst., 22(1), pp. 129–168. [CrossRef]
Chhabra, R. , 2016, “ Dynamical Reduction and Output-Tracking Control of the Lunar Exploration Light Rover (LELR),” 2016 IEEE, Aerospace Conference, Mar. 5–12, Big Sky, MT.
Bullo, F. , and Zefran, M. , 2002, “ On Mechanical Control Systems With Nonholonomic Constraints and Symmetries,” Syst. Control Lett., 45(2), pp. 133–143. [CrossRef]
Olfati-Saber, R. , 2001, “ Nonlinear Control of Underactuated Mechanical Systems With Application to Robotics and Aerospace Vehicles,” Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, MA.
Ferrario, C. , and Passerini, A. , 2000, “ Rolling Rigid Bodies and Forces of Constraint: An Application to Affine Nonholonomic Systems,” Meccanica, 35(5), pp. 433–442. [CrossRef]
Sun, W. , Wu, Y. Q. , and Sun, Z. Y. , 2014, “ Tracking Control Design for Nonholonomic Mechanical Systems With Affine Constraints,” Int. J. Autom. Comput., 11(3), pp. 328–333. [CrossRef]
Fassó, F. , and Sansonetto, N. , 2015, “ Conservation of Energy and Momenta in Nonholonomic Systems With Affine Constraints,” Regular Chaotic Dyn., 20(4), pp. 449–462. [CrossRef]
Noether, E. , 1918, “ Invariante Variationsprobleme, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen,” Mathematisch-Physikalische Klasse, pp. 235–257.
Marsden, J. E. , and Weinstein, A. , 1974. “ Reduction of Symplectic Manifolds With Symmetry,” Rep. Math. Phys., 5(1), pp. 121–130. [CrossRef]
Routh, E. J. , 1882, A Treatise on the Dynamics of a System of Rigid Bodies. With Numerous Examples: The Elementary Part, Macmillan, London.
Marsden, J. E. , 1992, Lectures on Mechanics, Cambridge University Press, New York.
Planas-Bielsa, V. , 2004, “ Point Reduction in Almost Symplectic Manifolds,” Rep. Math. Phys., 54(3), pp. 295–308. [CrossRef]
Marsden, J. E. , Misiolek, G. , and Ortega, J. P. , 2007, Hamiltonian Reduction by Stages, 1st ed., Springer-Verlag, Berlin.
Chhabra, R. , and Emami, M. R. , 2015, “ Symplectic Reduction of Holonomic Open-Chain Multi-Body Systems With Constant Momentum,” J. Geom. Phys., 89, pp. 82–110. [CrossRef]
Koon, W. S. , and Marsden, J. E. , 1998, “ Poisson Reduction for Nonholonomic Mechanical Systems With Symmetry,” Rep. Math. Phys., 42(1–2), pp. 101–134. [CrossRef]
Cendra, H. , Marsden, J. E. , and Ratiu, T. S. , 2001, Lagrangian Reduction by Stages, Vol. 722, American Mathematical Society, Providence, RI.
Marsden, J. E. , and Scheurle, J. , 1993, “ The Reduced Euler–Lagrange Equations,” Fields Inst. Commun., 1, pp. 139–164.
Marsden, J. E. , and Scheurle, J. , 1993, “ Lagrangian Reduction and the Double Spherical Pendulum,” Z. Angew. Math. Phys., 44(1), pp. 17–43. [CrossRef]
Chaplygin, S. , 2008, “ On the Theory of Motion of Nonholonomic Systems. The Reducing-Multiplier Theorem,” Regular Chaotic Dyn., 13(4), pp. 369–376 [Matematicheskiĭ sbornik, 28(1) (1911)]. [CrossRef]
Koiller, J. , 1992, “ Reduction of Some Classical Non-Holonomic Systems With Symmetry,” Arch. Ration. Mech. Anal., 118(2), pp. 113–148. [CrossRef]
van der Schaft, A. J. , and Maschke, B. M. , 1994, “ On the Hamiltonian Formulation of Nonholonomic Mechanical Systems,” Rep. Math. Phys., 34(2), pp. 225–233. [CrossRef]
Yoshimura, H. , and Marsden, J. E. , 2006, “ Dirac Structures in Lagrangian Mechanics Part II: Variational Structures,” J. Geom. Phys., 57(1), pp. 209–250. [CrossRef]
Cendra, H. , Marsden, J. E. , and Ratiu, T. S. , 2001, “ Geometric Mechanics, Lagrangian Reduction and Nonholonomic Systems,” Mathematics Unlimited-2001 and Beyond, Springer-Verlag, Berlin, pp. 221–273.
Chhabra, R. , and Emami, M. R. , 2014, “ Nonholonomic Dynamical Reduction of Open-Chain Multi-Body Systems: A Geometric Approach,” Mech. Mach. Theory, 82, pp. 231–255. [CrossRef]
Ohsawa, T. , Fernandez, O. E. , Bloch, A. M. , and Zenkov, D. V. , 2011, “ Nonholonomic Hamilton-Jacobi Theory Via Chaplygin Hamiltonization,” J. Geom. Phys., 61(8), pp. 1263–1291. [CrossRef]
Hochgerner, S. , and Garcia-Naranjo, L. , 2009, “ G-Chaplygin Systems With Internal Symmetries, Truncation, and an (Almost) Symplectic View of Chaplygin's Ball,” J. Geom. Mech., 1(1), pp. 35–53. [CrossRef]
Bates, L. , and Śniatycki, J. , 1993, “ Nonholonomic Reduction,” Rep. Math. Phys., 32(1), pp. 99–115. [CrossRef]
Cushman, R. , Kemppainen, D. , Śniatycki, J. , and Bates, L. , 1995, “ Geometry of Nonholonomic Constraints,” Rep. Math. Phys., 36(2/3), pp. 275–286. [CrossRef]
Cushman, R. , and Śniatycki, J. , 2002, “ Nonholonomic Reduction for Free and Proper Actions,” Reg. Chaotic Dyn., 7(1), pp. 61–72. [CrossRef]
Cushman, R. , Duistermaat, H. , and Śniatycki, J. , 2009, Geometry of Nonholonomically Constrained Systems, World Scientific Publishing Company, Singapore.
Śniatycki, J. , 1998, “ Nonholonomic Noether Theorem and Reduction of Symmetries,” Rep. Math. Phys., 42(1/2), pp. 5–23. [CrossRef]
Śniatycki, J. , 2001, “ Almost Poisson Spaces and Nonholonomic Singular Reduction,” Rep. Math. Phys., 48(1/2), pp. 235–248. [CrossRef]
Gay-Balmaz, F. , and Yoshimura, H. , 2015, “ Dirac Reduction for Nonholonomic Mechanical Systems and Semidirect Products,” Adv. Appl. Math., 63, pp. 131–213. [CrossRef]


Grahic Jump Location
Fig. 1

A rover on a rotating disk with angular velocity Σ



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In