Poisson schemes for Hamiltonian systems on Poisson manifolds
type: Item
instance of: scholarly article
title: Poisson schemes for Hamiltonian systems on Poisson manifolds
zbMATH Open document ID: 0807.58018
DOI: 10.1016/0898-1221(94)90081-7
author: Wenjie Zhu
author: Qin Meng Zhao
publication date: 1994-08-07T00:00:00Z
review text: At the beginning, the authors recall a more general kind of Hamiltonian systems, which can be obtained by expanding the canonical Poisson brackets to general Poisson brackets. The phase spaces of this kind of systems can be of odd or infinite dimension, so they can describe many more problems in dynamical systems. A Poisson bracket on \(\mathbb{R}^ n\) is defined as a binary operator \(\{\cdot ,\cdot\}\) on \(C^ \infty(\mathbb{R}^ n)\) by \(\{F,H\} = (\nabla F)^ T K(\nabla H)\), where \(K\) is any given antisymmetric \(n \times n\) matrix. The space \(\mathbb{R}^ n\) provided with the Poisson bracket is called a Poisson manifold and a mapping \(g : \mathbb{R}^ n \to \mathbb{R}^ n\) which preserves the Poisson bracket is a Poisson mapping. A Hamiltonian system on \(\mathbb{R}^ n\) is the following system of ODE: \(dz/dt = K \nabla H(z)\). The authors consider the linear case \(H(z) = z^ T Sz\), \(z \in \mathbb{R}^ n\), \(S\) is a symmetric constant matrix. For numerical computation of the phase flow a sequence of Poisson difference schemes based on the \((k,k)\)-order Padé approximations is obtained. A difference scheme regarded as a mapping is called a Poisson difference scheme if this mapping is a Poisson mapping. The paper is well written and organized.
Mathematics Subject Classification ID: 37J99
Mathematics Subject Classification ID: 65L06
Mathematics Subject Classification ID: 41A21
zbMATH DE Number: 612171
zbMATH Keywords: Padé approximation
zbMATH Keywords: implicit Runge-Kutta method
zbMATH Keywords: Hamiltonian systems
zbMATH Keywords: Poisson brackets
zbMATH Keywords: Poisson manifold
reviewed by: A. Klíč
MaRDI profile type: Publication
full work available at URL: 0898-1221(94)90081-7
OpenAlex ID: W2057618366
cites work: Q3809192
cites work: Q3729973
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cites work: Almost Poisson integration of rigid body systems
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cites work: Canonical difference scheme for the Hamiltonian equation
cites work: Q4291922
published in: Computers & Mathematics with Applications
description: scientific article; zbMATH DE number 612171