On a family of integrable systems on <i>S</i><sup>2</sup> with a cubic integral of motion

Tsiganov, A. V.

doi:10.1088/0305-4470/38/4/011

Cited by 13 publications

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“…For other separable natural bi-integrable systems from [23,34] we present Hamiltonians and functions g and h only. So, for the Goryachev-Chaplygin top [11,7] we have…”

Section: Case 2 -Systems With Cubic Integral Of Motionmentioning

confidence: 99%

On natural Poisson bivectors on the sphere

Tsiganov¹

2011

J. Phys. A: Math. Theor.

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We discuss the concept of natural Poisson bivectors, which allows us to consider the overwhelming majority of known integrable systems on the sphere in framework of bi-Hamiltonian geometry. IntroductionThe Hamilton-Jacobi theory seems to be one of the most powerful methods of investigation the dynamics of mechanical (holonomic and nonholonomic) and control systems. Besides its fundamental aspects such as its relation to the action integral and generating functions of symplectic maps, the theory is known to be very useful in integrating the Hamilton equations using the variables separation technique. The milestones of this technique include the works of Stäckel, Levi-Civita, Eisenhart, Woodhouse, Kalnins, Miller, Benenti and others. The majority of results was obtained for a very special class of integrable systems, important from the physical point of view, namely for the systems with quadratic in momenta integrals of motion. The Kowalevski, Chaplygin and Goryachev results on separation of variables for the systems with higher order integrals of motion missed out of this scheme.Bi-Hamiltonian structures can be seen as a dual formulation of integrability and separability, in the sense that they substitute a hierarchy of compatible Poisson structures to the hierarchy of functions in involution, which may be treated either as integrals of motion or as variables of separation for some dynamical system. The Eisenhart-Benenti theory was embedded into the bi-Hamiltonian set-up using the lifting of the conformal Killing tensor that lies at the heart of Benenti's construction [8,15]. The concept of natural Poisson bivectors allows us to generalize this construction and to study systems with quadratic and higher order integrals of motion in framework of a single theory [31].The aim of this note is to bring together all the known examples of natural Poisson bivectors on the sphere, because a good example is the best sermon. Some of these Poisson bivectors have been obtained and presented earlier in different coordinate systems and notations. Here we propose the unified description of this known and few new bivectors using so-called geodesic Π and potential Λ matrices [31]. In some sense we propose new form for the old content and believe that this unification is a first step to the geometric analysis of various natural systems on the sphere, which reveals what they have in common and indicates the most suitable strategy to obtain and to analyze their solutions.The corresponding integrable natural systems on two-dimensional unit sphere S 2 are related to rigid body dynamics. In order to describe these systems we will use the angular momentum vector J = (J 1 , J 2 , J 3 ) and the Poisson vector x = (x 1 , x 2 , x 3 ) in a moving frame of coordinates attached to the principal axes of inertia [4]. The Poisson brackets between these variablesmay be associated to the Lie-Poisson algebra of the three-dimensional Euclidean algebra e(3) with two Casimir elementsx k J k .(1.2)Below we always put C 2 = 0.

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“…For other separable natural bi-integrable systems from [23,34] we present Hamiltonians and functions g and h only. So, for the Goryachev-Chaplygin top [11,7] we have…”

Section: Case 2 -Systems With Cubic Integral Of Motionmentioning

confidence: 99%

On natural Poisson bivectors on the sphere

Tsiganov¹

2011

J. Phys. A: Math. Theor.

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“…In the Goryachev-Chaplygin case the integral also has degree 3 in the momenta (see § 2.4). Moreover, Tsiganov [22] presented a 4-parameter family of integrable cases on M 4 1,0 having integrals of degree 3 in the momenta. In that paper Tsiganov looked for a Hamiltonian and an additional first integral of the following form:…”

Section: Bifurcations Of Liouville Torimentioning

confidence: 99%

“…Here λ and g are related by means of the sixth equation of the system (22). We substitute the expression for H in (27):…”

Section: Bifurcation Diagram Of the Moment Mapmentioning

confidence: 99%

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Topology of the Liouville foliation on a 2-sphere in the Dullin-Matveev integrable case

Moskvin

2008

Sb. Math.

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The paper is concerned with the study of the topology of the Liouville foliations of the Dullin-Matveev integrable case. The critical point set of the Hamiltonian is found, the types of isoenergy surfaces are calculated, the non-degeneracy conditions are verified, the types of non-degenerate points of the Poisson action are determined, the moment map is investigated and the bifurcation diagram is constructed. A test for the Bott property is verified by numerical simulation. The indices of critical circles, the bifurcation types and the rough molecules are found. The rough Liouville classification of this integrable case is virtually accomplished as a result.Bibliography: 24 titles.

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“…Some special commutative subalgebras generated by quadratic polynomials on coefficients of T ij (λ) were considered in [8]. These subalgebras were associated with five integrable systems on S 2 with an additional integral of motion third order in the momenta.…”

Section: Generic Casementioning

confidence: 99%

A new integrable system onS²with the second integral quartic in the momenta

Tsiganov

2005

J. Phys. A: Math. Gen.

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On a family of integrable systems on S² with a cubic integral of motion

Cited by 13 publications

References 5 publications

On natural Poisson bivectors on the sphere

On natural Poisson bivectors on the sphere

Topology of the Liouville foliation on a 2-sphere in the Dullin-Matveev integrable case

A new integrable system onS²with the second integral quartic in the momenta

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On a family of integrable systems on S2 with a cubic integral of motion

Cited by 13 publications

References 5 publications

On natural Poisson bivectors on the sphere

On natural Poisson bivectors on the sphere

Topology of the Liouville foliation on a 2-sphere in the Dullin-Matveev integrable case

A new integrable system onS2with the second integral quartic in the momenta

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On a family of integrable systems on S² with a cubic integral of motion

A new integrable system onS²with the second integral quartic in the momenta