On generalized hamiltonian systems

Daizhan Cheng; Weimin Xue; Lizhi Liao; Dayong Cai

doi:10.1007/bf02669700

On generalized hamiltonian systems

Daizhan Cheng^*
, Weimin Xue
, Lizhi Liao
, Dayong Cai

^*Corresponding author for this work

Department of Mathematics

Research output: Contribution to journal › Journal article › peer-review

1 Citation (Scopus)

Abstract

The purpose of this paper is to explore an extension of some fundamental properties of the Hamiltonian systems to a more general case. We first extend symplectic group to a general Ngroup, G_N, and prove that it has certain similar properties. A particular property of G_N is that as a Lie group dim (G_N)≥1. Certain properties of its Lie-algebra gjv are investigated. The results obtained are used to investigate the structure-preserving systems, which generalize the property of symplectic form preserving of Hamiltonian system to a covariant tensor field preserving of certain dynamic systems. The results provide a theoretical benchmark of applying symplectic algorithm to a considerably larger class of structure-preserving systems.

Original language	English
Pages (from-to)	475-483
Number of pages	9
Journal	Acta Mathematicae Applicatae Sinica
Volume	17
Issue number	4
DOIs	https://doi.org/10.1007/bf02669700
Publication status	Published - 2001

User-Defined Keywords

Hamiltonian control systems
Hamiltonian systems
Symplectic algebra
Symplectic algorithm
Symplectic group

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10.1007/bf02669700

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@article{760f4a84a9464ef1a6945203d87703f5,

title = "On generalized hamiltonian systems",

abstract = "The purpose of this paper is to explore an extension of some fundamental properties of the Hamiltonian systems to a more general case. We first extend symplectic group to a general Ngroup, GN, and prove that it has certain similar properties. A particular property of GN is that as a Lie group dim (GN)≥1. Certain properties of its Lie-algebra gjv are investigated. The results obtained are used to investigate the structure-preserving systems, which generalize the property of symplectic form preserving of Hamiltonian system to a covariant tensor field preserving of certain dynamic systems. The results provide a theoretical benchmark of applying symplectic algorithm to a considerably larger class of structure-preserving systems.",

keywords = "Hamiltonian control systems, Hamiltonian systems, Symplectic algebra, Symplectic algorithm, Symplectic group",

author = "Daizhan Cheng and Weimin Xue and Lizhi Liao and Dayong Cai",

note = "Funding Information: Definition 1.1 \textbackslash{}[1,2\textbackslash{}]. Let M be a manifold, w E \textasciitilde{}2(M) is called a symplectic form on M, if w is nonsingular and closed. M with a symplectic form, (M,w), is called a symplectic manifold; We use \textasciitilde{}t2(M) for the set of skew-symmetric 2nd order covariant tensor fields, which are also called two-forms. Intuitively, it is a coordinate-depending skew-symmetric quadratic Received June 14, 1999. Revised June 28, 2000. * This research is partly supported by National Natural Science Foundation of China (No.G59837270, G1998020308) and the National Key Project of China.",

year = "2001",

doi = "10.1007/bf02669700",

language = "English",

volume = "17",

pages = "475--483",

journal = "Acta Mathematicae Applicatae Sinica",

issn = "0168-9673",

publisher = "Springer Verlag",

number = "4",

}

TY - JOUR

T1 - On generalized hamiltonian systems

AU - Cheng, Daizhan

AU - Xue, Weimin

AU - Liao, Lizhi

AU - Cai, Dayong

N1 - Funding Information: Definition 1.1 \[1,2\]. Let M be a manifold, w E ~2(M) is called a symplectic form on M, if w is nonsingular and closed. M with a symplectic form, (M,w), is called a symplectic manifold; We use ~t2(M) for the set of skew-symmetric 2nd order covariant tensor fields, which are also called two-forms. Intuitively, it is a coordinate-depending skew-symmetric quadratic Received June 14, 1999. Revised June 28, 2000. * This research is partly supported by National Natural Science Foundation of China (No.G59837270, G1998020308) and the National Key Project of China.

PY - 2001

Y1 - 2001

N2 - The purpose of this paper is to explore an extension of some fundamental properties of the Hamiltonian systems to a more general case. We first extend symplectic group to a general Ngroup, GN, and prove that it has certain similar properties. A particular property of GN is that as a Lie group dim (GN)≥1. Certain properties of its Lie-algebra gjv are investigated. The results obtained are used to investigate the structure-preserving systems, which generalize the property of symplectic form preserving of Hamiltonian system to a covariant tensor field preserving of certain dynamic systems. The results provide a theoretical benchmark of applying symplectic algorithm to a considerably larger class of structure-preserving systems.

AB - The purpose of this paper is to explore an extension of some fundamental properties of the Hamiltonian systems to a more general case. We first extend symplectic group to a general Ngroup, GN, and prove that it has certain similar properties. A particular property of GN is that as a Lie group dim (GN)≥1. Certain properties of its Lie-algebra gjv are investigated. The results obtained are used to investigate the structure-preserving systems, which generalize the property of symplectic form preserving of Hamiltonian system to a covariant tensor field preserving of certain dynamic systems. The results provide a theoretical benchmark of applying symplectic algorithm to a considerably larger class of structure-preserving systems.

KW - Hamiltonian control systems

KW - Hamiltonian systems

KW - Symplectic algebra

KW - Symplectic algorithm

KW - Symplectic group

UR - https://www.scopus.com/pages/publications/52549096505

U2 - 10.1007/bf02669700

DO - 10.1007/bf02669700

M3 - Journal article

AN - SCOPUS:52549096505

SN - 0168-9673

VL - 17

SP - 475

EP - 483

JO - Acta Mathematicae Applicatae Sinica

JF - Acta Mathematicae Applicatae Sinica

IS - 4

ER -

On generalized hamiltonian systems

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