Lie‐Poisson integrators: A Hamiltonian, variational approach

Ma, Zhanhua; Rowley, Clarence W.

doi:10.1002/nme.2812

Cited by 10 publications

(9 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Therefore, in the sequel we describe the geometric schemes that extend the coadjoint orbit preserving integration method for SO (3) [8,16,18]. Another possibility of constructing the structure-preserving algorithms is to follow the variational approach, see, for example [20,36], and references cited there.…”

Section: Coadjoint Modeling Of Rotational Dynamicsmentioning

confidence: 98%

Modelling and Integration Concepts of Multibody Systems on Lie Groups

Müller

Terze

2014

Computational Methods in Applied Sciences

View full text Add to dashboard Cite

Lie group integration schemes for multibody systems (MBS) are attractive as they provide a coordinate-free and thus singularity-free approach to MBS modeling. The Lie group setting also allows for developing integration schemes that preserve motion integrals and coadjoint orbits. Most of the recently proposed Lie group integration schemes are based on variants of generalized alpha Newmark schemes. In this chapter constrained MBS are modeled by a system of differentialalgebraic equations (DAE) on a configuration space being a subvariety of the Lie group SE(3) n . This is transformed to an index 1 DAE system that is integrated with Munthe-Kaas (MK) integration scheme. The chapter further addresses geometric integration schemes that preserve integrals of motion. In this context, a non-canonical Lie-group Störmer-Verlet integration scheme with direct SO(3) rotational update is presented. The method is 2nd order accurate and it is angular momentum preserving for a free-spinning body. Moreover, although being fully explicit, the method achieves excellent conservation of the angular momentum of a free rotational body and the motion integrals of the Lagrangian top. A higher-order coadjoint-preserving integration scheme on SO(3) is also presented. This method exactly preserves spatial angular momentum of a free body and it is particularly numerically efficient. A. Müller (B)

show abstract

Section: Coadjoint Modeling Of Rotational Dynamicsmentioning

confidence: 98%

Modelling and Integration Concepts of Multibody Systems on Lie Groups

Müller

Terze

2014

Computational Methods in Applied Sciences

View full text Add to dashboard Cite

show abstract

“…In the field of MBS geometric integration, special attention is devoted to structure preserving methods that exploit rich geometric structure of rigid body rotational dynamics (see [7,16,25,58,67,73,84,85,87,89,90,97,104,135,146,147,152] and references cited therein). To this end, rigid body rotational dynamics is studied most conveniently as Lie-Poisson system that is defined on so * (3) (the dual space of so (3)).…”

Section: Geometric Integration Of Mbs Models In Absolute Coordinatesmentioning

confidence: 99%

“…Actually, Lie-Poisson system can be regarded as a reduced system resulting from Lie-Poisson reduction of a canonical Hamiltonian system [97,98]. It is linked to Euler-Poincare system (reduced Lagrangian being defined on pertinent Lie algebra) via reduced Legendre transform.…”

Section: Geometric Integration Of Mbs Models In Absolute Coordinatesmentioning

confidence: 99%

Geometric methods and formulations in computational multibody system dynamics

Müller

Terze

2016

Acta Mech

View full text Add to dashboard Cite

Multibody systems are dynamical systems characterized by intrinsic symmetries and invariants. Geometric mechanics deals with the mathematical modeling of such systems and has proven to be a valuable tool providing insights into the dynamics of mechanical systems, from a theoretical as well as from a computational point of view. Modeling multibody systems, comprising rigid and flexible members, as dynamical systems on manifolds, and Lie groups in particular, leads to frame-invariant and computationally advantageous formulations. In the last decade, such formulations and corresponding algorithms are becoming increasingly used in various areas of computational dynamics providing the conceptual and computational framework for multibody, coupled, and multiphysics systems, and their nonlinear control. The geometric setting, furthermore, gives rise to geometric numerical integration schemes that are designed to preserve the intrinsic structure and invariants of dynamical systems. These naturally avoid the long-standing problem of parameterization singularities and also deliver the necessary accuracy as well as a long-term stability of numerical solutions. The current intensive research in these areas documents the relevance and potential for geometric methods in general and in particular for multibody system dynamics. This paper provides an exhaustive summary of the development in the last decade, and a panoramic overview of the current state of knowledge in the field.

show abstract

“…Hamiltonian variational principles have been developed by Oh [1997] in the context of Floer homology and by Novikov [1982] for Morse theory (see also Cendra and Marsden [1987]). On the numerical front, geometrical numerical integration of Hamiltonian systems was described in Brown [2006], Ma and Rowley [2010] and Leok and Zhang [2011], but all of these references assume that the underlying symplectic manifold is exact. For non-exact symplectic forms (e.g.…”

Section: Background and Historical Overviewmentioning

confidence: 99%

A Novel Formulation of Point Vortex Dynamics on the Sphere: Geometrical and Numerical Aspects

Vankerschaver

Leok

2013

J Nonlinear Sci

View full text Add to dashboard Cite

In this paper, we present a novel Lagrangian formulation of the equations of motion for point vortices on the unit 2-sphere. We show first that no linear Lagrangian formulation exists directly on the 2-sphere but that a Lagrangian may be constructed by pulling back the dynamics to the 3-sphere by means of the Hopf fibration. We then use the isomorphism of the 3-sphere with the Lie group SU (2) to derive a variational Lie group integrator for point vortices which is symplectic, second-order, and preserves the unit-length constraint. At the end of the paper, we compare our integrator with classical fourth-order Runge-Kutta, the second-order midpoint method, and a standard Lie group Munthe-Kaas method.

show abstract

Lie‐Poisson integrators: A Hamiltonian, variational approach

Cited by 10 publications

References 12 publications

Modelling and Integration Concepts of Multibody Systems on Lie Groups

Modelling and Integration Concepts of Multibody Systems on Lie Groups

Geometric methods and formulations in computational multibody system dynamics

A Novel Formulation of Point Vortex Dynamics on the Sphere: Geometrical and Numerical Aspects

Contact Info

Product

Resources

About