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Time integration and discrete Hamiltonian systems. (English) Zbl 0866.58030

The paper deals with numerical integration schemes for Hamiltonian systems with symmetry. It is desirable that numerical time-integration schemes preserve physically meaningful integrals of the underlying system. These types of integrators are usually referred to as conserving integrators and are the subject of this investigation.
Let \((P,\Omega)\) be a symplectic space with \(P\) open in \(\mathbb{R}^m\) and \(H:P\to \mathbb{R}\) any smooth function. Consider the Hamiltonian differential equation (1): \(\dot z= X_H (z)\), where the Hamiltonian vector field \(X_H\) is assumed to be smooth. With respect to equation (1), the numerical scheme (2): \(z_{n+1}- z_n=h \widetilde X_H (z_n,z_{n+1})\) is constructed in such a way that it inherits an arbitrary integral from the underlying system. In (2), \(h>0\) is the time step and \(\widetilde X_H: P\times P\to \mathbb{R}^m\) is a given smooth map.
The author further defines discrete derivative, discrete Hamiltonian vector field and discrete bracket, which is used to define integrals for discrete Hamiltonian systems. Namely the ones with symmetry are investigated within the context of discrete dynamical systems.
This important paper is clearly written and well organized.
Reviewer: A.Klíč (Praha)

MSC:

37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
37C80 Symmetries, equivariant dynamical systems (MSC2010)
65D30 Numerical integration
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