A new integrable case of the motion of the 4-dimensional rigid body

DOI10.1007/BF01205938zbMath0606.58029MaRDI QIDQ1084725

A. G. Reyman, Michaael A. Semenov-Tian-Shansky

Publication date: 1986

Published in: Communications in Mathematical Physics (Search for Journal in Brave)

zbMATH Keywords

affine Lie algebrasLax equationsexceptional Lie algebrastwisted loop algebradynamics of rigid bodiesclassical mechanics on Riemannian symmetric pairsintegrable topsKostant-Adler-Symes commutativity theoremmagnetohydrodynamical models of pulsar rotationmotion of interacting tops

Mathematics Subject Classification ID

Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) (37K10) Free motion of a rigid body (70E15) Infinite-dimensional Lie (super)algebras (17B65) Applications of Lie groups to the sciences; explicit representations (22E70) Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests (37J35) Hydrodynamic and hydromagnetic problems in astronomy and astrophysics (85A30)

Related Items (27)

Explicit integration of the system of invariant relations for the case of M. Adler and P. van Moerbeke ⋮ Regularization of Toda lattices by Hamiltonian reduction ⋮ Orbit interpretation of Hamiltonian systems of the type of an anharmonic oscillator ⋮ The Toda flows preserving small cells of the flag variety \(G/B\) and Kazhdan's \(x_{0}\)-grading ⋮ Argument shift method and sectional operators: applications to differential geometry ⋮ Integrability of the periodic KM system ⋮ Quantum tops as examples of commuting differential operators ⋮ On the complete integrability of nonlinear dynamical systems on functional manifolds within the gradient-holonomic approach ⋮ Factorization in a torus and Riemann-Hilbert problems ⋮ Bi-Hamiltonian structure of the \(qd\) algorithm and new discretizations of the Toda lattice ⋮ A family of integrable symplectic standard-like maps related to symmetric spaces ⋮ On a Poisson reduction for Gel'fand-Zakhare\-vich manifolds ⋮ Automorphisms of \(\text{sl}(2)\) and classical integrable systems ⋮ Dynamical \(r\)-matrices for the constrained Harry-Dym flows ⋮ Matrix Riemann--Hilbert problems and factorization on Riemann surfaces ⋮ Spectral parameter dependent Lax pairs for systems of Calogero-Moser type ⋮ Hamiltonian chaos in nonlinear optical polarization dynamics ⋮ Integrable quadratic Hamiltonians with a linear Lie-Poisson bracket ⋮ A discrete nonlinear Schrödinger-type hierarchy, its finite-dimensional reduction analysis, and the numerical integration scheme ⋮ Generalized tops and moment maps to loop algebras ⋮ The Adler-van Moerbeke integrable case. Visualization of bifurcations of Liouville tori ⋮ The integrable case of Adler-van Moerbeke. Discriminant set and bifurcation diagram ⋮ The Kowalewski top 99 years later: a Lax pair, generalizations and explicit solutions ⋮ Classical M. A. Buhl problem, its Pfeiffer-Sato solutions, and the classical Lagrange-d'Alembert principle for the integrable heavenly-type nonlinear equations ⋮ Hamiltonian and gradient structures in the Toda flows ⋮ Dynamical systems on quantum tori Lie algebras ⋮ On matrix differential equations and abstract FG algorithm

Cites Work

• Integrable Hamiltonian systems connected with graded Lie algebras
• What is a classical r-matrix?
• New integrable problem of classical mechanics
• Integrability of two interacting \(n\)-dimensional rigid bodies
• Completely integrable systems, Euclidean Lie algebras, and curves
• Note on the integration of Euler's equations of the dynamics of an \(n\)-dimensional rigid body
• Semidirect Products and Reduction in Mechanics
• Geodesic flow on SO(4) and the intersection of quadrics
• Euler-Poisson Equations on Lie Algebras and the N-Dimensional Heavy Rigid Body

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