Symplectic Methods for Conservative Multibody Systems

Besides preserving the energy, the ow of a conservative multibody system possesses important geometric (symplectic) invariants. Symplectic discretization schemes that mimic the corresponding feature of the true ow have been shown to be eeective alternatives to standard methods for many conservative problems. For systems of rigid bodies, the development of such schemes can be complicated or costly to implement, depending on the choice of problem formulation. In this article, we demonstrate that a special formulation of the multibody system (based on a particle representation) together with a symplectic discretization for constrained problems borrowed from molecular dynamics ooers an eecient alternative to standard approaches. Numerical experiments illustrating this approach are described. 1. Introduction In this paper, we consider eecient numerical integrators for systems of rigid bodies interconnected by various types of mechanical joints and subject to the forces of nature. Important applications include \dynamic models" 17] of undamped and mildly damped mechanisms and manipulators 38, 23]. Symplectic schemes have been found to be eeective for diverse problems such as astronomical many-body systems 36] and molecular dynamics simulations 22]. We are particularly interested in two aspects of symplectic integrators: rst, all other things being equal, they appear to provide realistic simulations on longer time intervals than nonsymplectic schemes, and second, the condition for being symplectic is connected to other important properties such as norm-preservation and angular momentum-preservation. All symplectic schemes are necessarily volume preserving. In some cases, as in our example of spinning top (Section 6), conservation of integrals proves critical to successful numerical simulation. In standard treatments of classical mechanics 2], a rigid body is viewed as a collection of point masses with xed interparticle distances. The natural description of the body as a system of cartesian coordinates for the locations of the point masses is typically replaced by a minimal coordinate representation, for example based on the introduction of quaternions. The idea of using cartesian (\natural") coordinates for the simulation of multibody systems was rst discussed by 37, 6], with the aim of improving eeciency in real-time multibody simulation. Our approach is based on a procedure we term particulation in which a given rigid body is replaced by a system of point particles while preserving its dynamical properties. Some tools for constructing such particle representations were rst presented by Edward Routh in his classic 1905 treatise 32]. We describe a somewhat generalized version of his procedure, discuss the imposition of …