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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

Professor
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.

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Grahic Jump Location
Fig. 1

A rover on a rotating disk with angular velocity Σ

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