Poisson schemes for Hamiltonian systems on Poisson manifolds
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.
scientific article; zbMATH DE number 612171