On the linearization of Hamiltonian systems on poisson manifolds

Yu M. Vorob'ev

doi:10.1007/s11006-005-0129-5

On the linearization of Hamiltonian systems on poisson manifolds

Yu M. Vorob'ev

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

Abstract

The linearization of a Hamiltonian system on a Poisson manifold at a given (singular) symplectic leaf gives a dynamical system on the normal bundle of the leaf, which is called the first variation system. We show that the first variation system admits a compatible Hamiltonian structure if there exists a transversal to the leaf which is invariant with respect to the flow of the original system. In the case where the transverse Lie algebra of the symplectic leaf is semisimple, this condition is also necessary.

Original language	English
Pages (from-to)	297-303
Number of pages	7
Journal	Mathematical Notes
Volume	78
Issue number	3-4
DOIs	https://doi.org/10.1007/s11006-005-0129-5
State	Published - Sep 2005

Bibliographical note

Funding Information:
The author is very grateful to M. V. Karasev and R. Flores Espinoza for helpful discussions of various aspects of the present work. This research was supported in part by CONACYT under grant no. 35212-E.

Keywords

First variation system
Hamiltonian system
Hamiltonian vector field
Linearization
Normal bundle
Poisson bracket
Poisson coupling

Access to Document

10.1007/s11006-005-0129-5

Cite this

@article{abe513b0df454086be9f16a7217b7b98,

title = "On the linearization of Hamiltonian systems on poisson manifolds",

abstract = "The linearization of a Hamiltonian system on a Poisson manifold at a given (singular) symplectic leaf gives a dynamical system on the normal bundle of the leaf, which is called the first variation system. We show that the first variation system admits a compatible Hamiltonian structure if there exists a transversal to the leaf which is invariant with respect to the flow of the original system. In the case where the transverse Lie algebra of the symplectic leaf is semisimple, this condition is also necessary.",

keywords = "First variation system, Hamiltonian system, Hamiltonian vector field, Linearization, Normal bundle, Poisson bracket, Poisson coupling",

author = "Vorob'ev, {Yu M.}",

note = "Funding Information: The author is very grateful to M. V. Karasev and R. Flores Espinoza for helpful discussions of various aspects of the present work. This research was supported in part by CONACYT under grant no. 35212-E.",

year = "2005",

month = sep,

doi = "10.1007/s11006-005-0129-5",

language = "Ingl{\'e}s",

volume = "78",

pages = "297--303",

journal = "Mathematical Notes",

issn = "0001-4346",

publisher = "Pleiades Publishing",

number = "3-4",

}

TY - JOUR

T1 - On the linearization of Hamiltonian systems on poisson manifolds

AU - Vorob'ev, Yu M.

N1 - Funding Information: The author is very grateful to M. V. Karasev and R. Flores Espinoza for helpful discussions of various aspects of the present work. This research was supported in part by CONACYT under grant no. 35212-E.

PY - 2005/9

Y1 - 2005/9

N2 - The linearization of a Hamiltonian system on a Poisson manifold at a given (singular) symplectic leaf gives a dynamical system on the normal bundle of the leaf, which is called the first variation system. We show that the first variation system admits a compatible Hamiltonian structure if there exists a transversal to the leaf which is invariant with respect to the flow of the original system. In the case where the transverse Lie algebra of the symplectic leaf is semisimple, this condition is also necessary.

AB - The linearization of a Hamiltonian system on a Poisson manifold at a given (singular) symplectic leaf gives a dynamical system on the normal bundle of the leaf, which is called the first variation system. We show that the first variation system admits a compatible Hamiltonian structure if there exists a transversal to the leaf which is invariant with respect to the flow of the original system. In the case where the transverse Lie algebra of the symplectic leaf is semisimple, this condition is also necessary.

KW - First variation system

KW - Hamiltonian system

KW - Hamiltonian vector field

KW - Linearization

KW - Normal bundle

KW - Poisson bracket

KW - Poisson coupling

UR - http://www.scopus.com/inward/record.url?scp=27144455712&partnerID=8YFLogxK

U2 - 10.1007/s11006-005-0129-5

DO - 10.1007/s11006-005-0129-5

M3 - Artículo

SN - 0001-4346

VL - 78

SP - 297

EP - 303

JO - Mathematical Notes

JF - Mathematical Notes

IS - 3-4

ER -

On the linearization of Hamiltonian systems on poisson manifolds

Abstract

Bibliographical note

Keywords

Access to Document

Other files and links

Fingerprint

Cite this