Extended Integrability and Bi-Hamiltonian Systems

Bogoyavlenskij, Oleg I.

doi:10.1007/s002200050412

Cited by 106 publications

(87 citation statements)

References 21 publications

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“…As pointed out in particular by Bogoyavlenskij [2], but see also [9], quasi-periodicity of a flow can be (semilocally) linked to the presence of a number of first integrals and of a complementary number of commuting dynamical symmetries which preserve these first integrals. Specifically, if on a manifold M and for some n < d = dim M there are • a submersion F : M → R d−n with compact and connected fibers, and • n everywhere linearly independent and pairwise commuting vector fields Y 1 , .…”

Section: The Phase Map and The Integrals Of Motionmentioning

confidence: 99%

Geometry of Invariant Tori of Certain Integrable Systems with Symmetry and an Application to a Nonholonomic System

Fassò

Giacobbe

2007

SIGMA

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Abstract. Bifibrations, in symplectic geometry called also dual pairs, play a relevant role in the theory of superintegrable Hamiltonian systems. We prove the existence of an analogous bifibrated geometry in dynamical systems with a symmetry group such that the reduced dynamics is periodic. The integrability of such systems has been proven by M. Field and J. Hermans with a reconstruction technique. We apply the result to the nonholonomic system of a ball rolling on a surface of revolution.

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Section: The Phase Map and The Integrals Of Motionmentioning

confidence: 99%

Geometry of Invariant Tori of Certain Integrable Systems with Symmetry and an Application to a Nonholonomic System

Fassò

Giacobbe

2007

SIGMA

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“…In [25], Bogoyavlenskij noted that a locally Hamiltonian system that describes the motion of an electron on a two-dimensional torus T 2 under the action of an electromagnetic field in the limit where the ratio of the small Shevchenko Kyiv National University, Kyiv.…”

Section: Introductionmentioning

confidence: 99%

Theorem on perturbation of coisotropic invariant tori of locally Hamiltonian systems and its applications

Loveikin

Parasyuk

2005

Nonlinear Oscill

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We study the problem of perturbations of quasiperiodic motions on coisotropic invariant tori in a class of locally Hamiltonian systems. We prove a general KAM-theorem on the perturbation of coisotropic invariant tori for locally Hamiltonian systems. As applications of this theorem, we consider the motion of an electron on a two-dimensional torus under the action of an electromagnetic field and extend results concerning the bifurcation of a Cantor set of coisotropic invariant tori to the case of locally Hamiltonian systems.

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“…For instance this is the case of superintegrable Hamiltonian systems when n is even, due to the Liouville-Arnold Theorem, see for more details [1,2,4,12] and the references therein.…”

Section: Consider a Cmentioning

confidence: 99%

The Completely Integrable Differential Systems are Essentially Linear Differential Systems

2015

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Abstract. Letẋ = f (x) be a C k autonomous differential system with k ∈ N ∪ {∞, ω} defined in an open subset Ω of R n . Assume that the systemẋ = f (x) is C r completely integrable, i.e. there exist n − 1 functionally independent first integrals of class C r with 2 ≤ r ≤ k. If the divergence of systemẋ = f (x) is non-identically zero, then any Jacobian multiplier is functionally independent of the n − 1 first integrals. Moreover the systemẋ = f (x) is C r−1 orbitally equivalent to the linear differential systemẏ = y in a full Lebesgue measure subset of Ω. For Darboux and polynomial integrable polynomial differential systems we characterize their type of Jacobian multipliers.

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Extended Integrability and Bi-Hamiltonian Systems

Cited by 106 publications

References 21 publications

Geometry of Invariant Tori of Certain Integrable Systems with Symmetry and an Application to a Nonholonomic System

Geometry of Invariant Tori of Certain Integrable Systems with Symmetry and an Application to a Nonholonomic System

Theorem on perturbation of coisotropic invariant tori of locally Hamiltonian systems and its applications

The Completely Integrable Differential Systems are Essentially Linear Differential Systems

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