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Volume 16, Issue 2
Variations on a Theme by Euler

Kang Feng & Daoliu Wang

J. Comp. Math., 16 (1998), pp. 97-106.

Published online: 1998-04

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  • Abstract

The oldest and simplest difference scheme is the explicit Euler method. Usually, it is not symplectic for general Hamiltonian systems. It is interesting to ask: Under what conditions of Hamiltonians, the explicit Euler method becomes symplectic? In this paper, we give the class of Hamiltonians for which systems the explicit Euler method is symplectic. In fact, in these cases, the explicit Euler method is really the phase flow of the systems, therefore symplectic. Most of important Hamiltonian systems can be decomposed as the summation of these simple systems. Then composition of the Euler method acting on these systems yields a symplectic method, also explicit. These systems are called symplectically separable. Classical separable Hamiltonian systems are symplectically separable. Especially, we prove that any polynomial Hamiltonian is symplectically separable.

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@Article{JCM-16-97, author = {Feng , Kang and Wang , Daoliu}, title = {Variations on a Theme by Euler}, journal = {Journal of Computational Mathematics}, year = {1998}, volume = {16}, number = {2}, pages = {97--106}, abstract = {

The oldest and simplest difference scheme is the explicit Euler method. Usually, it is not symplectic for general Hamiltonian systems. It is interesting to ask: Under what conditions of Hamiltonians, the explicit Euler method becomes symplectic? In this paper, we give the class of Hamiltonians for which systems the explicit Euler method is symplectic. In fact, in these cases, the explicit Euler method is really the phase flow of the systems, therefore symplectic. Most of important Hamiltonian systems can be decomposed as the summation of these simple systems. Then composition of the Euler method acting on these systems yields a symplectic method, also explicit. These systems are called symplectically separable. Classical separable Hamiltonian systems are symplectically separable. Especially, we prove that any polynomial Hamiltonian is symplectically separable.

}, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/9144.html} }
TY - JOUR T1 - Variations on a Theme by Euler AU - Feng , Kang AU - Wang , Daoliu JO - Journal of Computational Mathematics VL - 2 SP - 97 EP - 106 PY - 1998 DA - 1998/04 SN - 16 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/9144.html KW - Hamiltonian systems, symplectic difference schemes, explicit Euler method, nilpotent, symplectically separable. AB -

The oldest and simplest difference scheme is the explicit Euler method. Usually, it is not symplectic for general Hamiltonian systems. It is interesting to ask: Under what conditions of Hamiltonians, the explicit Euler method becomes symplectic? In this paper, we give the class of Hamiltonians for which systems the explicit Euler method is symplectic. In fact, in these cases, the explicit Euler method is really the phase flow of the systems, therefore symplectic. Most of important Hamiltonian systems can be decomposed as the summation of these simple systems. Then composition of the Euler method acting on these systems yields a symplectic method, also explicit. These systems are called symplectically separable. Classical separable Hamiltonian systems are symplectically separable. Especially, we prove that any polynomial Hamiltonian is symplectically separable.

Kang Feng & Daoliu Wang. (1970). Variations on a Theme by Euler. Journal of Computational Mathematics. 16 (2). 97-106. doi:
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