Hamiltonian Systems near Relative Equilibria

Roberts, Mark; Wulff, Claudia; Lamb, Jeroen S. W.

doi:10.1006/jdeq.2001.4045

Cited by 55 publications

(115 citation statements)

References 45 publications

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“…The full dynamics correspond to Euclidean extensions, given by the reconstruction maps, of the reduced dynamics. Corollaries 4.4 and 4.5 are consistent with the general theory of Euclidean invariant Hamiltonian systems (see [13]). …”

Section: )supporting

confidence: 74%

Interaction of two charges in a uniform magnetic field: I. Planar problem

Pinheiro

MacKay

2006

Nonlinearity

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Abstract. The interaction of two charges moving in R 2 in a magnetic field B can be formulated as a Hamiltonian system with 4 degrees of freedom. Assuming that the magnetic field is uniform and the interaction potential has rotational symmetry we reduce this Hamiltonian system to one with 2 degrees of freedom; for certain values of the conserved quantities and choices of parameters, we obtain an integrable system. Furthermore, when the interaction potential is of Coulomb type, we prove that, for suitable regime of parameters, there are invariant subsets on which this system contains a suspension of a subshift of finite type. This implies non-integrability for this system with a Coulomb type interaction. Explicit knowledge of the reconstruction map and a dynamical analysis of the reduced Hamiltonian systems are the tools we use in order to give a description for the various types of dynamical behaviours in this system: from periodic to quasiperiodic and chaotic orbits, from bounded to unbounded motion.

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Section: )supporting

confidence: 74%

Interaction of two charges in a uniform magnetic field: I. Planar problem

Pinheiro

MacKay

2006

Nonlinearity

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“…The idea exploited in [8] and [9] consists in taking the pullback by ϕ of the Hamiltonian vector field X h to Y and then lift it to G × m * × N . In these two references the differential equations for the flow of a particular choice of the lifted vector field are obtained, providing a general framework to study the local dynamics of X h near G · z.…”

Section: The Bundle Equations Around Relative Equilibriamentioning

confidence: 99%

“…Here the symbol P denotes the canonical projection onto the subspace indicated by the subscript relative to a splitting of some vector space, which will be clear from the context. As shown in [8,9], these are the differential equations for the flow of the unique local vector field X on G × m * × N which is a lift of ϕ * X h ∈ X(Y ) and satisfies the condition…”

Section: The Bundle Equations Around Relative Equilibriamentioning

confidence: 99%

“…The organization of the article is as follows: Section 2 contains some generalities about Hamiltonian actions and Hamiltonian relative equilibria, including a brief review of the construction of the Marle-Guillemin-Sternberg form (MGS). Then we introduce the so-called bundle equations as introduced in [8] and [9], which can be thought as Hamilton's equation for an invariant Hamiltonian function on the tubular model provided by the MGS form. Next, we show the main novelty of our approach which consists in including the explicit role of the isotropy of the group action into the bundle equations, obtaining our main tool, the system of equations (11) (12) and (13).…”

Section: Introductionmentioning

confidence: 99%

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A sufficient condition for the existence of Hamiltonian bifurcations with continuous isotropy

Montaldi

Rodriguez-Olmos²

2013

Acta Math Vietnam.

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Abstract. We present a framework for the study of the local qualitative dynamics of equivariant Hamiltonian flows specially designed for points in phase space with nontrivial isotropy. This is based on the classical construction of structure-preserving tubular neighborhoods for Hamiltonian Lie group actions on symplectic manifolds. This framework is applied to obtaining concrete and testable conditions guaranteeing the existence of bifurcations from symmetric branches of Hamiltonian relative equilibria.

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“…These equations are known in the Hamiltonian context under the name of Reconstruction equations [O98,OR02] or Bundle equations [RWL99].…”

Section: Introductionmentioning

confidence: 99%

The Symplectic Slice Theorem

Ortega

Raţiu

2004

Momentum Maps and Hamiltonian Reduction

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We provide a model for an open invariant neighborhood of any orbit in a symplectic manifold endowed with a canonical proper symmetry. Our results generalize the constructions of Marle [Mar84,Mar85] and Guillemin and Sternberg [GS84] for canonical symmetries that have an associated momentum map. In these papers the momentum map played a crucial role in the construction of the tubular model. The present work shows that in the construction of the tubular model it can be used the so called Chu map [Chu75] instead, which exists for any canonical action, unlike the momentum map. Hamilton's equations for any invariant Hamiltonian function take on a particularly simple form in these tubular variables. As an application we will find situations, that we will call tubewise Hamiltonian, in which the existence of a standard momentum map in invariant neighborhoods is guaranteed.

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Hamiltonian Systems near Relative Equilibria

Cited by 55 publications

References 45 publications

Interaction of two charges in a uniform magnetic field: I. Planar problem

Interaction of two charges in a uniform magnetic field: I. Planar problem

A sufficient condition for the existence of Hamiltonian bifurcations with continuous isotropy

The Symplectic Slice Theorem

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