The Kowalewski and Hénon-Heiles motions as Manakov geodesic flows on SO(4) - a two-dimensional family of Lax pairs

DOI10.1007/BF01223242zbMath0647.58022OpenAlexW2032713492MaRDI QIDQ1104612

Mark Adler, Pierre van Moerbeke

Publication date: 1987

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

Full work available at URL: https://doi.org/10.1007/bf01223242

zbMATH Keywords

homogeneous manifoldsHénon-Heiles systemalgebraic integrability of mechanical systemsManakov geodesic flow

Mathematics Subject Classification ID

Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) (37K10) Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests (37J35) Geodesic flows in symplectic geometry and contact geometry (53D25) Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) (37D40)

Related Items (22)

The intersection of four quadrics in \(P^ 6\), Abelian surfaces and their moduli ⋮ The Painlevé-Kowalevski and Poly-Painlevé Tests for Integrability ⋮ Reduction of divisors and the Clebsch system ⋮ On isogenies among certain abelian surfaces ⋮ Unnamed Item ⋮ Lie point symmetries and first integrals: The Kowalevski top ⋮ Completely integrable systems: Jacobi's heritage ⋮ Riemann-Roch theorem and Kodaira-Serre duality ⋮ Elliptic solutions and blow-up in an integrable Hénon–Heiles system ⋮ Unnamed Item ⋮ Generic hyperelliptic Prym varieties in a generalized Hénon-Heiles system ⋮ Algebraic Description of Jacobians Isogeneous to Certain Prym Varieties with Polarization (1,2) ⋮ Kowalewski top and complex Lie algebras ⋮ Linearising two-dimensional integrable systems and the construction of action-angle variables ⋮ The generalized Hénon-Heiles system, abelian surfaces and algebraic complete integrability ⋮ Bifurcations of invariant manifolds in the generalized Hénon-Heiles system ⋮ Generalized tops and moment maps to loop algebras ⋮ Unnamed Item ⋮ The Kowalewski top 99 years later: a Lax pair, generalizations and explicit solutions ⋮ Noncanonical time transformations relating finite-dimensional integrable systems ⋮ Reductions of self-dual Yang-Mills fields and classical systems ⋮ The Kowalevski top as a reduction of a Hamiltonian system on $\mathfrak {sp}(4, \mathbb {R})^*$

Cites Work

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• Geodesic flow on \(SO(4)\) and Abelian surfaces
• The intersection of four quadrics in \(P^ 6\), Abelian surfaces and their moduli
• A Lax pair for Kowalevski's top
• The Goryachev-Chaplygin top and the Toda lattice
• Isospectral Hamiltonian flows in finite and infinite dimensions. I: Generalized Moser systems and moment maps into loop algebras
• Affine parts of abelian surfaces as complete intersections of four quadrics
• Lax representation for the systems of S. Kovalevskaya type
• Some remarks on the integrability of the equations of motion of a rigid body in an ideal fluid
• The full Geometry of Kowalewski's top and (1, 2)-abelian surfaces
• Geodesic flow on SO(4) and the intersection of quadrics
• Linearizing Flows and a Cohomological Interpretation of Lax Equations
• The prime radical for Ω-groups

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