Symplectic topology of integrable Hamiltonian systems, I: Arnold-Liouville with singularities (original) (raw)
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Symplectic Topology of Integrable Hamiltonian Systems, II: Topological Classification
International Journal of Computer Mathematics, 2000
The classical Arnold-Liouville theorem describes the geometry of an integrable Hamiltonian system near a regular level set of the moment map. Our results describe it near a nondegenerate singular level set: a tubular neighborhood of a connected singular nondegenerate level set, after a normal finite covering, admits a non-complete system of action-angle functions (the number of action functions is equal to the rank of the moment map), and it can be decomposed topologically, together with the associated singular Lagrangian foliation, to a direct product of simplest (codimension 1 and codimension 2) singularities. These results are essential for the global topological study of integrable Hamiltonian systems.
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Symplectic Singularities and Solvable Hamiltonian Mappings
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GEOMETRY OF INTEGRABLE NON-HAMILTONIAN SYSTEMS
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Symplectic bifurcation theory for integrable systems
2011
This paper develops a symplectic bifurcation theory for integrable systems in dimension four. We prove that if an integrable system has no hyperbolic singularities and its bifurcation diagram has no vertical tangencies, then the fibers of the induced singular Lagrangian fibration are connected. The image of this singular Lagrangian fibration is, up to smooth deformations, a planar region bounded by the graphs of two continuous functions. The bifurcation diagram consists of the boundary points in this image plus a countable collection of rank zero singularities, which are contained in the interior of the image. Because it recently has become clear to the mathematics and mathematical physics communities that the bifurcation diagram of an integrable system provides the best framework to study symplectic invariants, this paper provides a setting for studying quantization questions, and spectral theory of quantum integrable systems.
Reports on Mathematical Physics, 2000
A symplectic theory approach is developed for solving the problem of algebraicanalytical construction of integral submanifold imbedding mapping for integrable via the abelian Liouville-Arnold theorem Hamiltonian systems on canonically symplectic phase spaces. The related Picard-Fuchs type equations are derived for the first time straightforwardly, making use of a method based on generalized Francoise-Galissot-Reeb differential-geometric results. The relationships between toruslike compact integral submanifolds of a LiouvilleArnold integrable Hamiltonian system and solutions to corresponding Picard-Fuchs type equations is stated. 1. General setting 1.1. Our main object of study will be differential systems of vector fields on the cotangent phase space M2" = T*(IW"), n E Z+, endowed with the canonical symplectic structure wc2) E A2(M2n), where ~(~1 = d(pr*c&')), and (Y(l) := (p, dq) = &dq,, j=1 (1.1) is the canonical l-form on the base space W", lifted naturally to the space A1 (Mzn), (q, p) E M2" are canonical coordinates on T*(B!"), pr : T*(IRY)-+ IR is the canonical projection, and (., .) is the usual scalar product in IIXn. Assume further that there is also given a Lie subgroup G (not necessarily compact), acting symplectically via the mapping cp : G x M2" + M2" on M2n, generating a Lie
Haantjes Manifolds of Classical Integrable Systems
2016
A general theory of classical integrable systems is proposed, based on the geometry of the Haantjes tensor. We introduce the class of symplectic-Haantjes manifolds (or ωH manifold), as the natural setting where the notion of integrability can be formulated. We prove that the existence of suitable Haantjes structures is a necessary and sufficient condition for a Hamiltonian system to be integrable in the Liouville-Arnold sense. We also prove theorems ensuring the existence of a large family of completely integrable systems, constructed starting from a prescribed Haantjes structure. Furthermore, we propose a novel approach to the theory of separation of variables, intimately related to the geometry of Haantjes manifolds. A special family of coordinates, that we shall call the Darboux-Haantjes coordinates, will be introduced. They are constructed from the Haantjes structure associated with an integrable system, and allow the additive separation of variables of the Hamilton-Jacobi equat...
On singularities of Hamiltonian mappings
Geometry and topology of caustics, 2008
The notion of an implicit Hamiltonian system-an isotropic mapping H : M → (T M,ω) into the tangent bundle endowed with the symplectic structure defined by canonical morphism between tangent and cotangent bundles of M-is studied. The corank one singularities of such systems are classified. Their transversality conditions in the 1-jet space of isotropic mappings are described and the corresponding symplectically invariant algebras of Hamiltonian generating functions are calculated. We have F * ω = 0 andF = π • F,ω = β −1 (dθ).