The Lagrange bitop on so (4) × so (4) and geometry of the Prym varieties

Dragović, Vladimir; Gajić, Borislav

doi:10.1353/ajm.2004.0035

Cited by 10 publications

(69 citation statements)

References 27 publications

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“…is the Hamiltonian function of the Lagrange bitop. The Lagrange bitop is a complete integrable system of a heavy rigid body on so(4) × so(4) defined in [18] and studied in details in [19]. Moreover, the Lagrange bitop is a bi-Hamiltonian system.…”

Section: 2mentioning

confidence: 99%

“…In other words, the systems (7.10), restricted to (7.14), is an example of integrable isoholomorhic system ( [19], see Remark 8.2 given below).…”

Section: Partial Reduction Of Rigid Body Systemsmentioning

confidence: 99%

“…Since the orbit O(χ) ∼ = Gr + (4, 2) is defined with invariants, the cotangent bundle of the Grassmannian Gr + (4, 2) is given by the constraints: The equations (8.2) are special case of the equations of the completely symmetric Lagrange bitop. The Lagrange bitop is defined in [18] and studied in details in [19]. Thus, the integration procedures given in [18,19] can be applied to the considered system.…”

Section: Integration Of the Magnetic Pendulum On Gr + (4 2)mentioning

confidence: 99%

“…Generalization of this Lax pair in fourdimensional case led to construction of a new integrable rigid-body system in [18], called the Lagrange bitop. Algebro-geometric integration procedure of the Lagrange bitop has been performed in [19]. It brought to the discovery of a new class of integrable systems, which was named isoholonomic systems in [19].…”

mentioning

confidence: 99%

“…Algebro-geometric integration procedure of the Lagrange bitop has been performed in [19]. It brought to the discovery of a new class of integrable systems, which was named isoholonomic systems in [19]. Higher dimensional generalizations of the classical rigid-body Hess-Appel'rot systems have been constructed in [20,21] as certain perturbations of the isoholonomic integrable systems.…”

mentioning

confidence: 99%

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Systems of Hess–appel'rot Type and Zhukovskii Property

Dragović

Gajić²,

Jovanović

2009

Int. J. Geom. Methods Mod. Phys.

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We start with a review of a class of systems with invariant relations, so called systems of Hess-Appel'rot type that generalizes the classical Hess-Appel'rot rigid body case. The systems of Hess-Appel'rot type have remarkable property: there exists a pair of compatible Poisson structures, such that a system is certain Hamiltonian perturbation of an integrable bi-Hamiltonian system. The invariant relations are Casimir functions of the second structure. The systems of Hess-Appel'rot type carry an interesting combination of both integrable and non-integrable properties.Further, following integrable line, we study partial reductions and systems having what we call the Zhukovskii property: These are Hamiltonian systems on a symplectic manifold M with actions of two groups G and K; the systems are assumed to be Kinvariant and to have invariant relation Φ = 0 given by the momentum mapping of the G-action, admitting two types of reductions, a reduction to the Poisson manifold P = M/K and a partial reduction to the symplectic manifold N 0 = Φ −1 (0)/G; final and crucial assumption is that the partially reduced system to N 0 is completely integrable. We prove that the Zhukovskii property is a quite general characteristic of systems of Hess-Appel'rot type. The partial reduction neglects the most interesting and challenging part of the dynamics of the systems of Hess-Appel'rot type -the non-integrable part, some analysis of which may be seen as a reconstruction problem.We show that an integrable system, the magnetic pendulum on the oriented Grassmannian Gr + (n, 2) has a natural interpretation within Zhukovskii property and that it is equivalent to a partial reduction of certain system of Hess-Appel'rot type. We perform a classical and algebro-geometric integration of the system in dimension four, as an example of a known isoholomorphic system -the Lagrange bitop.The paper presents a lot of examples of systems of Hess-Appel'rot type, giving an additional argument in favor of further study of this class of systems.

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Section: 2mentioning

confidence: 99%

“…In other words, the systems (7.10), restricted to (7.14), is an example of integrable isoholomorhic system ( [19], see Remark 8.2 given below).…”

Section: Partial Reduction Of Rigid Body Systemsmentioning

confidence: 99%

Section: Integration Of the Magnetic Pendulum On Gr + (4 2)mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Systems of Hess–appel'rot Type and Zhukovskii Property

Dragović

Gajić²,

Jovanović

2009

Int. J. Geom. Methods Mod. Phys.

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Geodesic flows and Neumann systems on Stiefel varieties: geometry and integrability

Fedorov¹,

Jovanović²

2010

Math. Z.

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We study integrable geodesic flows on Stiefel varieties Vn,r = SO(n)/SO(n−r) given by the Euclidean, normal (standard), Manakov-type, and Einstein metrics. We also consider natural generalizations of the Neumann systems on Vn,r with the above metrics and proves their integrability in the non-commutative sense by presenting compatible Poisson brackets on (T * Vn,r)/SO(r). Various reductions of the latter systems are described, in particular, the generalized Neumann system on an oriented Grassmannian Gn,r and on a sphere S n−1 in presence of Yang-Mills fields or a magnetic monopole field.Apart from the known Lax pair for generalized Neumann systems, an alternative (dual) Lax pair is presented, which enables one to formulate a generalization of the Chasles theorem relating the trajectories of the systems and common linear spaces tangent to confocal quadrics. Additionally, several extensions are considered: the generalized Neumann system on the complex Stiefel variety Wn,r = U (n)/U (n − r), the matrix analogs of the double and coupled Neumann systems.

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Systems of Hess-Appel’rot Type

Dragović

Gajić

2006

Commun. Math. Phys.

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We construct higher-dimensional generalizations of the classical Hess-Appel'rot rigid body system. We give a Lax pair with a spectral parameter leading to an algebro-geometric integration of this new class of systems, which is closely related to the integration of the Lagrange bitop performed by us recently and uses Mumford relation for theta divisors of double unramified coverings. Based on the basic properties satisfied by such a class of systems related to bi-Poisson structure, quasi-homogeneity, and conditions on the Kowalevski exponents, we suggest an axiomatic approach leading to what we call the "class of systems of Hess-Appel'rot type".

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The Lagrange bitop on so (4) × so (4) and geometry of the Prym varieties

Cited by 10 publications

References 27 publications

Systems of Hess–appel'rot Type and Zhukovskii Property

Systems of Hess–appel'rot Type and Zhukovskii Property

Geodesic flows and Neumann systems on Stiefel varieties: geometry and integrability

Systems of Hess-Appel’rot Type

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