Geometry of Integrable non-Hamiltonian Systems

Zung, Nguyen Tien

doi:10.1007/978-3-319-33503-2_3

Cited by 11 publications

(12 citation statements)

References 63 publications

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“…In particular, there is still a topological decomposition into direct products of simpler singularities, as will be seen in §5. One can also talk about the (real) toric degree of these singularities (see [25] and references therein for the notion of toric degree), a topic that we will not discuss in this paper.…”

Section: Introductionmentioning

confidence: 99%

Singular fibres of the Gelfand–Cetlin system on MANI( n )*

Bouloc

Miranda

Zung

2018

Phil. Trans. R. Soc. A.

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In this paper, we show that every singular fibre of the Gelfand-Cetlin system on co-adjoint orbits of unitary groups is a smooth isotropic submanifold which is diffeomorphic to a two-stage quotient of a compact Lie group by free actions of two other compact Lie groups. In many cases, these singular fibres can be shown to be homogeneous spaces or even diffeomorphic to compact Lie groups. We also give a combinatorial formula for computing the dimensions of all singular fibres, and give a detailed description of these singular fibres in many cases, including the so-called (multi-)diamond singularities. These (multi-)diamond singular fibres are degenerate for the Gelfand-Cetlin system, but they are Lagrangian submanifolds diffeomorphic to direct products of special unitary groups and tori. Our methods of study are based on different ideas involving complex ellipsoids, Lie groupoids and also general ideas coming from the theory of singularities of integrable Hamiltonian systems.This article is part of the theme issue 'Finite dimensional integrable systems: new trends and methods'.

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Section: Introductionmentioning

confidence: 99%

Singular fibres of the Gelfand–Cetlin system on MANI( n )*

Bouloc

Miranda

Zung

2018

Phil. Trans. R. Soc. A.

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“…In connection with our results, we would like to mention the theorem of Chaperon on smooth equivalence of formally equivalent weakly hyperbolic systems [2], and its recent application to smooth geometric linearization of some classes of integrable non-Hamiltonian systems by Jiang [6]. We believe that Chaperon's techniques will be a key element in our smooth linearization problem, see also [14].…”

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confidence: 73%

Orbital Linearization of Smooth Completely Integrable Vector Fields

Zung¹

2017

SIGMA

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The main purpose of this paper is to prove the smooth local orbital linearization theorem for smooth vector fields which admit a complete set of first integrals near a nondegenerate singular point. The main tools used in the proof of this theorem are the formal orbital linearization theorem for formal integrable vector fields, the blowing-up method, and the Sternberg-Chen isomorphism theorem for formally-equivalent smooth hyperbolic vector fields

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“…We begin by recalling the notion of non-Hamiltonian integrable systems ( [1,27]; for an Action-Angle Theorem in this setting, see [28]). A vector field X on an ndimensional manifold M is called integrable if there exist p vector fields X 1 = X, X 2 , .…”

Section: Cross Products Of Integrable Systemsmentioning

confidence: 99%

“…Singularities of integrable Hamiltonian systems have been studied by many authors. More recently, there has been interest in formulating a theory of singularities of integrable non-Hamiltonian systems (see [27] and references therein). A detailed study of singularities of concrete integrable systems on configuration spaces of planar linkages would help the development of this theory.…”

Section: Singularities Of Integrable Systems On Linkage Spacesmentioning

confidence: 99%

“…Thus, by studying natural integrable systems on configuration spaces of planar linkages, we can obtain interesting integrable non-Hamiltonian systems on any closed manifold. The theory of integrable non-Hamiltonian systems (in the sense of Bogoyavlensky [2]) is relatively new compared to integrable Hamiltonian systems, especially when it comes to their topological and geometric properties (see [27] and references therein), so this huge class of integrable non-Hamiltonian systems coming from planar linkages will be very useful for the development of the theory. We summarize below the main results of the paper.…”

Section: Introductionmentioning

confidence: 99%

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Integrable systems in planar robotics

Ratiu,

Zung

2019

Preprint

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The main purpose of this paper is to investigate commuting flows and integrable systems on the configuration spaces of planar linkages. Our study leads to the definition of a natural volume form on each configuration space of planar linkages, the notion of cross products of integrable systems, and also the notion of multi-Nambu integrable systems. The first integrals of our systems are functions of Bott-Morse type, which may be used to study the topology of configuration spaces.

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Geometry of Integrable non-Hamiltonian Systems

Cited by 11 publications

References 63 publications

Singular fibres of the Gelfand–Cetlin system on MANI( n )*

Singular fibres of the Gelfand–Cetlin system on MANI( n )*

Orbital Linearization of Smooth Completely Integrable Vector Fields

Integrable systems in planar robotics

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