Hamiltonian Structure and Integrability

Fuchssteiner, Benno

doi:10.1016/s0076-5392(08)62801-5

Cited by 14 publications

(30 citation statements)

References 15 publications

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“…How the results of this paper are applied in the classical situation of Hamiltonian systems see [4]. The main message of the results of the present paper is that compatibility is more a property of homomorphisms with respect to bilinear structures than a property connected to vector fields.…”

Section: Introductionmentioning

confidence: 88%

Compatibility in Abstract Algebraic Structures

Fuchssteiner

1997

Algebraic Aspects of Integrable Systems

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Compatible Hamiltonian pairs play a crucial role in the structure theory of integrable systems. In this paper we consider the question of how much of the structure given by compatibility is bound to the situation of hamiltonian dynamic systems and how much of that can be transferred to a complete abstract situation where the algebraic structures under consideration are given by bilinear maps on some module over a commutative ring. Under suitable modification of the corresponding definitions, it turns out that notions like, compatible, hereditary, invariance and Virasoro algebra may be transferred to the general abstract setup. Thus the same methods being so successful in the area of integrable systems, may be applied to generate suitable abelian algebras and hierarchies in very general algebraic structures.

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Section: Introductionmentioning

confidence: 88%

Compatibility in Abstract Algebraic Structures

Fuchssteiner

1997

Algebraic Aspects of Integrable Systems

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“…More generally, it usually suffices for E, F to be locally convex Hausdorff rather than Banach vector spaces and f to be Hadamard-differentiable, cf. Fuchssteiner (1992a). This allows for a variety of manifolds, e.g., modeled over the space of rapidly decreasing functions S or equipped with some inductive limit topology.…”

Section: Manifolds Flows and Integrabilitymentioning

confidence: 99%

“…Indeed, the basic properties that lead to infinitely many conserved quantities and symmetries for (2) are usually stated in purely algebraic terms (Zakharov and Faddeev, 1971;Fuchssteiner, 1992a) 5 and a sound analytical foundation is given only later by supplying M with a suitable topology. Observe that equations of the form (2) cover evolving physical systems ranging from simple pendulum d 2 dt 2 ϕ + sin ϕ = 0, i.e.…”

Section: Manifolds Flows and Integrabilitymentioning

confidence: 99%

“…Hamiltonian systems have successfully been generalized from finite to infinite dimensional manifolds (Chernoff and Marsden, 1974;Choquet-Bruhat et al, 1991) and proven to be the key to integrability of many difficult nonlinear partial differential equations (Zakharov and Faddeev, 1971;Fuchssteiner, 1992a). We recall that if the generator K of an evolutionary equation…”

Section: Introductionmentioning

confidence: 99%

“…Theorem 4.3, relying on on a result in combinatorial algebra (Ma and Racine, 1990), formally justifies this identification. In fact, Fuchssteiner (1992a) revealed the basic properties of usual Hamiltonian systems that relate symmetries to conserved quantities and asserted the complete integrability of so many important nonlinear flows (Zakharov and Faddeev, 1971;Gardner, 1971;Fuchssteiner et al, 1997) to be expressible in algebraic terms only, too. Our main result (Section 5) proves each such abstract generator of Heisenberg's dynamics to be Hamiltonian in this generalized algebraic sense.…”

Section: Introductionmentioning

confidence: 99%

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Nonlinear Reformulation of Heisenberg's Dynamics

Ziegler

Fuchssteiner

2005

Int J Theor Phys

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A structural similarity between Classical Mechanics (CM) and Quantum Mechanics (QM) was revealed by P.A.M. Dirac in terms of Lie Algebras: while in CM the dynamics is determined by the Lie algebra of Poisson brackets on the manifold of scalar fields for classical position/momentum observables Q/P , QM evolves (in Heisenberg's picture) according to the formally similar Lie algebra of commutator brackets of the corresponding operators:where QP − PQ = ih. A further common framework for comparing CM and QM is the category of symplectic manifolds. Other than previous authors, this paper considers phase space of Heisenberg's picture, i.e., the manifold of pairs of operator observables (Q, P) satisfying commutation relation. On a sufficiently high algebraic level of abstraction-which we believe to be of interest on its own-it turns out that this approach leads to a truly nonlinear yet Hamiltonian reformulation of QM evolution.

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The relationship between the conservation laws and multi‐Hamiltonian structures of the Kundu equation

Zhang

Gongye

2022

Math Methods in App Sciences

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By the Lagrangian multiplier and constraint variational derivative, a relationship between conserved quantities and multi-Hamiltonian structures is built. Using the relation, a method is founded to prove the infinitedimensional Liouville integrability of evolution equations with continuous variables. As the application, the conservation laws of the Kundu equation are first obtained. Its conserved quantities are deduced for comparing by Fokas' method different from the method used in the existed literature. The integrability of the equation is proved through taking the conservation laws as a starting point.

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Hamiltonian Structure and Integrability

Cited by 14 publications

References 15 publications

Compatibility in Abstract Algebraic Structures

Compatibility in Abstract Algebraic Structures

Nonlinear Reformulation of Heisenberg's Dynamics

The relationship between the conservation laws and multi‐Hamiltonian structures of the Kundu equation

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