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21
Mechanical Integrators Derived from a Discrete Variational Principle
"... Many numerical integrators for mechanical system simulation are created by using discrete algorithms to approximate the continuous equations of motion. In this paper, we present a procedure to construct timestepping algorithms that approximate the flow of continuous ODE's for mechanical systems by ..."
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Cited by 60 (12 self)
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Many numerical integrators for mechanical system simulation are created by using discrete algorithms to approximate the continuous equations of motion. In this paper, we present a procedure to construct timestepping algorithms that approximate the flow of continuous ODE's for mechanical systems by discretizing Hamilton's principle rather than the equations of motion. The discrete equations share similarities to the continuous equations by preserving invariants, including the symplectic form and the momentum map. We girst present a formulation of discrete mechanics along with a discrete variational principle. We then show that the resulting equations of motion preserve the symplectic form and that this formulation of mechanics leads to conservation laws from a discrete version of Noether's theorem. We then use the discrete mechanics formulation to develop a procedure for constructing mechanical integrators for continuous Lagrangian systems. We apply the construction procedure to the rigid body and the double spherical pendulum to demonstrate numerical properties of the integrators.
Geometric numerical integration illustrated by the StörmerVerlet method
, 2003
"... The subject of geometric numerical integration deals with numerical integrators that preserve geometric properties of the flow of a differential equation, and it explains how structure preservation leads to improved longtime behaviour. This article illustrates concepts and results of geometric nume ..."
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Cited by 27 (4 self)
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The subject of geometric numerical integration deals with numerical integrators that preserve geometric properties of the flow of a differential equation, and it explains how structure preservation leads to improved longtime behaviour. This article illustrates concepts and results of geometric numerical integration on the important example of the Störmer–Verlet method. It thus presents a crosssection of the recent monograph by the authors, enriched by some additional material. After an introduction to the Newton–Störmer–Verlet–leapfrog method and its various interpretations, there follows a discussion of geometric properties: reversibility, symplecticity, volume preservation, and conservation of first integrals. The extension to Hamiltonian systems on manifolds is also described. The theoretical foundation relies on a backward error analysis, which translates the geometric properties of the method into the structure of a modified differential equation, whose flow is nearly identical to the numerical method. Combined with results from perturbation theory, this explains the excellent longtime behaviour of the method: longtime energy conservation, linear error growth and preservation of invariant tori in nearintegrable systems, a discrete virial theorem, and preservation of adiabatic invariants.
Hamilton–Pontryagin integrators on Lie groups
, 2007
"... Abstract In this paper, structurepreserving timeintegrators for rigid bodytype mechanical ..."
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Cited by 17 (6 self)
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Abstract In this paper, structurepreserving timeintegrators for rigid bodytype mechanical
Geometric Integrators for ODEs
 J. Phys. A
, 2006
"... Abstract. Geometric integration is the numerical integration of a differential equation, while preserving one or more of its “geometric ” properties exactly, i.e. to within roundoff error. Many of these geometric properties are of crucial importance in physical applications: preservation of energy, ..."
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Cited by 17 (5 self)
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Abstract. Geometric integration is the numerical integration of a differential equation, while preserving one or more of its “geometric ” properties exactly, i.e. to within roundoff error. Many of these geometric properties are of crucial importance in physical applications: preservation of energy, momentum, angular momentum, phase space volume, symmetries, timereversal symmetry, symplectic structure and dissipation are examples. In this paper we present a survey of geometric numerical integration methods for ordinary differential equations. Our aim has been to make the review of use for both the novice and the more experienced practitioner interested in the new developments and directions of the past decade. To this end, the reader who is interested in reading up on detailed technicalities will be provided with numerous signposts to the relevant literature. Geometric Integrators for ODEs 2 1.
A Symplectic Integrator for Riemannian Manifolds
, 1996
"... this article we show how such an ambient Riemannian metric can be used to frame the popular "leapfrog" method in the category of Riemannian manifolds, thus creating new intrinsic methods applicable to mechanical systems with configuration spaces as general as homogeneous spaces of semisimple Lie gro ..."
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Cited by 14 (2 self)
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this article we show how such an ambient Riemannian metric can be used to frame the popular "leapfrog" method in the category of Riemannian manifolds, thus creating new intrinsic methods applicable to mechanical systems with configuration spaces as general as homogeneous spaces of semisimple Lie groups. This new method, shown in Figure (1), is implicit, second order, timereversing, symplectic and respects those symmetries of the system which are also isometries of the ambient metric. For arbitrary potential energy, but where the kinetic energy metric of the mechanical system is proportional to the ambient metric, our method is explicit, and is equivalent to the splitting method corresponding to the splitting of the Hamiltonian into kinetic and potential energy. 2. Context and Notation
Structure Preservation For Constrained Dynamics With Super Partitioned Additive RungeKutta Methods
 SIAM J. Sci. Comput
, 1998
"... A broad class of partitioned differential equations with possible algebraic constraints is considered, including Hamiltonian and mechanical systems with holonomic constraints. For mechanical systems a formulation eliminating the Coriolis forces and closely related to the EulerLagrange equations is ..."
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Cited by 13 (9 self)
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A broad class of partitioned differential equations with possible algebraic constraints is considered, including Hamiltonian and mechanical systems with holonomic constraints. For mechanical systems a formulation eliminating the Coriolis forces and closely related to the EulerLagrange equations is presented. A new class of integrators is defined: the super partitioned additive RungeKutta (SPARK) methods. This class is based on the partitioning of the system into different variables and on the splitting of the differential equations into different terms. A linear stability and convergence analysis of these methods is given. SPARK methods allowing the direct preservation of certain properties are characterized. Different structures and invariants are considered: the manifold of constraints, symplecticness, reversibility, contractivity, dilatation, energy, momentum, and quadratic invariants. With respect to linear stability and structurepreservation, the class of sstage Lobatto IIIABCC* SPARK methods is of special interest. Controllable numerical damping can be introduced by the use of additional parameters. Some issues related to the implementation of a reversible variable stepsize strategy are discussed.
Symplectic Methods For Conservative Multibody Systems
 Fields Institute Communications
, 1993
"... . Besides preserving the energy, the flow of a conservative multibody system possesses important geometric (symplectic) invariants. Symplectic discretization schemes that mimic the corresponding feature of the true flow have been shown to be effective alternatives to standard methods for many conser ..."
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Cited by 11 (4 self)
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. Besides preserving the energy, the flow of a conservative multibody system possesses important geometric (symplectic) invariants. Symplectic discretization schemes that mimic the corresponding feature of the true flow have been shown to be effective 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 offers an efficient alternative to standard approaches. Numerical experiments illustrating this approach are described. 1. Introduction In this paper, we consider efficient numerical integrators for systems of rigid bodies interconnected by various types of mechanical joints and subject to the forces of natur...
The symmetric representation of the rigid body equations and their discretization
 141–71 FEDEROV Y N 2005 INTEGRABLE FLOWS AND BACKLUND TRANSFORMATIONS ON EXTENDED STIEFEL VARIETIES WITH APPLICATION TO THE EULER TOP ON THE LIE GROUP SO(3) PREPRINT NLIN.SI/0505045 AQ2 GELFAND I M AND FOMIN S V 2000 CALCULUS OF VARIATIONS TRANSLATED BY R
, 1998
"... This paper analyzes continuous and discrete versions of the generalized rigid body equations and the role of these equations in numerical analysis, optimal control and integrable Hamiltonian systems. In particular, we present a symmetric representation of the rigid body equations on the Cartesian pr ..."
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Cited by 11 (6 self)
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This paper analyzes continuous and discrete versions of the generalized rigid body equations and the role of these equations in numerical analysis, optimal control and integrable Hamiltonian systems. In particular, we present a symmetric representation of the rigid body equations on the Cartesian product SO(n) × SO(n) and study its associated symplectic structure. We describe the relationship of these ideas with the MoserVeselov theory of discrete integrable systems and with the theory of variational symplectic integrators. Preliminary work on the ideas discussed in the present paper may be found in Bloch, Crouch, Marsden and Ratiu [1998].
What kinds of dynamics are there? Lie pseudogroups, dynamical systems, and geometric integration
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Geometric integration algorithms on homogeneous manifolds
 Foundations of Computational Mathematics
, 2002
"... Given an ordinary differential equation on a homogeneous manifold, one can construct a “geometric integrator ” by determining a compatible ordinary differential equation on the associated Lie group, using a Lie group integration scheme to construct a discrete time approximation of the solution curve ..."
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Cited by 10 (3 self)
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Given an ordinary differential equation on a homogeneous manifold, one can construct a “geometric integrator ” by determining a compatible ordinary differential equation on the associated Lie group, using a Lie group integration scheme to construct a discrete time approximation of the solution curves in the group, and then mapping the discrete trajectories onto the homogeneous manifold using the group action. If the points of the manifold have continuous isotropy, a vector field on the manifold determines a continuous family of vector fields on the group, typically with distinct discretizations. If sufficient isotropy is present, an appropriate choice of vector field can yield improved capture of key features of the original system. In particular, if the algebra of the group is “full”, then the order of accuracy of orbit capture (i.e. approximation of trajectories modulo time reparametrization) within a specified family of integration schemes can be increased by an appropriate choice of isotropy element. We illustrate the approach developed here with comparisons of several integration schemes for the reduced rigid body equations on the sphere. 1 Introduction. Geometric integration techniques have become increasingly popular in the modern approach to numerical