Theory of Tensor Invariants of Integrable Hamiltonian Systems. II. Theorem on Symmetries and Its Applications

The theorem on symmetries is proved that states that a Liouville-integrable Hamiltonian system is non-degene\-rate in Kolmogorov's sense and has compact invariant submanifolds if and only if the corresponding Lie algebra of symmetries is abelian. The theorem on symmetries has applications to the characterization problem, to the integrable hierarchies problem, to the necessary conditions for the strong dynamical compatibility problem, and to the problem on master symmetries. The invariant necessary conditions for the non-degenerate C-integrability in Kolmogorov's sense of a given dynamical system V are derived. It is proved that the C-integrable Hamiltonian system is non-degenerate in the iso-energetic sense if and only if the corresponding Lie algebra of the iso-energetic conformal symmetries is abelian. An extended concept of integrability of Hamiltonian systems on the symplectic manifolds M ⁿ , n= 2k, is introduced. The concept of integrability describes the Hamiltonian systems that have quasi-periodic dynamics on tori or on toroidal cylinders of an arbitrary dimension . This concept includes, as a particular case, all Hamiltonian systems that are integrable in Liouville's classical sense, for which . The A-B-C-cohomologies are introduced for dynamical systems on smooth manifolds. Received: 16 January 1996 / Accepted: 3 July 1996     

四维相空间中非中心势动力学系统的Poisson 结构和Casimir 函数

将非中心势动力学系统的运动微分方程写成Ermakov形式，得到Ermakov不变量. 运用Hamilton理论，把Ermakov不变量当作Hamiltonian 函数，在四维相空间中建立了非中心势动力学系统的Poisson 结构。结果表明：此Poisson 结构是一退化的结构，而系统具有四个不变量，即Hamiltonian 函数，Ermakov不变量及两个Casimir函数。     

Third and fourth order invariants for one-dimensional time-dependent classical systems

The construction of invariants up to fourth order in velocities has been carried out for one-dimensional, time-dependent classical dynamical systems. While the exact results are recovered for the first and second order integrable systems, the results for the third and fourth order invariants are expressed in terms of nonlinearpotential equations. Noticing the separability of the potential in space and time variables these nonlinear equations are reduced to a tractable form. A possible solution for the third order case suggests a new integrable systemV(q, t) ∼t ^−4/3 q ^1/2. Alexander von Humboldt-Stiftung Fellow, on leave from the Department of Physics, Ramjas College (University of Delhi), Delhi 110 007, India.     

The Finite-Dimensional Moser Type Reduction of Modified Boussinesq and Super-Korteweg-de Vries Hamiltonian Systems via the Gradient-Holonomic Algorithm and Dual Moment Maps. Part I

Abstract

The Moser type reductions of modified Boussinessq and super-Korteweg-de Vries equations upon the finite-dimensional invariant subspaces of solutions are considered. For the Hamiltonian and Liouville integrable finite-dimensional dynamical systems concerned with the invariant subspaces, the Lax representations via the dual moment maps into some deformed loop algebras and the finite hierarchies of conservation laws are obtained. A supergeneralization of the Neumann dynamical system is presented.     

Poisson Involutions, Spin Calogero-Moser Systems Associated with Symmetric Lie Subalgebras and the Symmetric Space Spin Ruijsenaars-Schneider Models

We develop a general scheme to construct integrable systems starting from realizations in symmetric coboundary dynamical Lie algebroids and symmetric coboundary dynamical Poisson groupoids. The method is based on the successive use of Dirac reduction and Poisson reduction. Then we show that certain spin Calogero-Moser systems associated with symmetric Lie subalgebras can be studied in this fashion. We also consider some spin-generalized Ruisjenaars-Schneider equations which correspond to the N-soliton solutions of affine Toda field theory. In this case, we show how the equations are obtained from the Dirac reduction of some Hamiltonian system on a symmetric coboundary dynamical Poisson groupoid.     

Quantum and classical spin clusters: disappearance of quantum numbers and Hamiltonian chaos

We present a direct link between manifestations of classical Hamiltonian chaos and quantum nonintegrability effects as they occur in quantum invariants. In integrable classical Hamiltonian systems, analytic invariants (integrals of the motion) can be constructed numerically by means of time averages of dynamical variables over phase-space trajectories, whereas in near-integrable models such time averages yield nonanalytic invariants with qualitatively different properties. Translated into quantum mechanics, the invariants obtained from time averages of dynamical variables in energy eigenstates provide a topographical map of the plane of quantized actions (quantum numbers) with properties which again depend sensitively on whether or not the classical integrability condition is satisfied. The most conspicuous indicator of quantum chaos is the disappearance of quantum numbers, a phenomenon directly related to the breakdown of invariant tori in the classical phase flow. All results are for a system consisting of two exchange-coupled spins with biaxial exchange and single-site anisotropy, a system with a nontrivial integrability condition.     

Commutative partially integrable systems on Poisson manifolds

We show that, as distinct from completely integrable Hamiltonian systems, a commutative partially integrable system admits different compatible Poisson structures on a phase manifold that are related by a recursion operator. The existence of action–angle coordinates around an invariant submanifold of such a partially integrable system is proved.     

Topological Invariants in Braid Theory

Many invariants of knots and links have their counterparts in braid theory. Often, these invariants are most easily calculated using braids. A braid is a set of n strings stretching between two parallel planes. This review demonstrates how integrals over the braid path can yield topological invariants. The simplest such invariant is the winding number – the net number of times two strings in a braid wrap about each other. But other, higher-order invariants exist. The mathematical literature on these invariants usually employs techniques from algebraic topology that may be unfamiliar to physicists and mathematicians in other disciplines. The primary goal of this paper is to introduce higher-order invariants using only elementary differential geometry.Some of the higher-order quantities can be found directly by searching for closed one-forms. However, the Kontsevich integral provides a more general route. This integral gives a formal sum of all finite order topological invariants. We describe the Kontsevich integral, and prove that it is invariant to deformations of the braid.Some of the higher-order invariants can be used to generate Hamiltonian dynamics of n particles in the plane. The invariants are expressed as complex numbers; but only the real part gives interesting topological information. Rather than ignoring the imaginary part, we can use it as a Hamiltonian. For n = 2, this will be the Hamiltonian for point vortex motion in the plane. The Hamiltonian for n = 3 generates more complicated motions.     

Integrable Substructure in a Korteweg Capillarity Model. A Karman-Tsien Type Constitutive Relation

A classical Korteweg capillarity system with a Karman-Tsien type (κ, ρ) constitutive relation is shown, via a Madelung transformation and use of invariants of motion, to admit integrable Hamiltonian subsystems.     

Morita Invariant Properties of Twisted Poisson Manifolds

In this letter, we study some invariants of integrable twisted Poisson manifolds under Morita equivalence. We show that two twisted symplectic manifolds are Morita equivalent if and only if their fundamental groups are isomorphic. Moreover, the first cohomology group of twisted Poisson manifolds is invariant under Morita equivalence. Finally, we prove that if two twisted Poisson manifolds are Morita equivalent, then their representation categories are equivalent.      

A Class of Integrable Spin Calogero-Moser Systems

 We introduce a class of spin Calogero-Moser systems associated with root systems of simple Lie algebras and give the associated Lax representations (with spectral parameter) and fundamental Poisson bracket relations. The associated integrable models (called integrable spin Calogero-Moser systems in the paper) and their Lax pairs are then obtained via Poisson reduction and gauge transformations. For Lie algebras of A _n -type, this new class of integrable systems includes the usual Calogero-Moser systems as subsystems. Our method is guided by a general framework which we develop here using dynamical Lie algebroids. Received: 19 October 2001 / Accepted: 7 June 2002 Published online: 21 October 2002 RID="*" ID="*" Research partially supported by NSF grant DMS00-72171.     

Solvability of F 4 Quantum Integrable Systems

It is shown that the F ₄ rational and trigonometric integrable systems are exactly solvable for arbitrary values of the coupling constants. Their spectra are found explicitly while eigenfunctions are obtained by pure algebraic means. For both systems new variables are introduced in which the Hamiltonian has an algebraic form being also (block)-triangular. These variables are a certain invariants of the F ₄ Weyl group. Both Hamiltonians preserve the same (minimal) flag of spaces of polynomials, which is found explicitly.     

Gravity from Lie Algebroid Morphisms

Inspired by the Poisson Sigma Model and its relation to 2d gravity, we consider models governing morphisms from T to any Lie algebroid E, where is regarded as a d-dimensional spacetime manifold. We address the question of minimal conditions to be placed on a bilinear expression in the 1-form fields, S^ij(X)A_iAj, so as to permit an interpretation as a metric on . This becomes a simple compatibility condition of the E-tensor S with the chosen Lie algebroid structure on E. For the standard Lie algebroid E=TM the additional structure is identified with a Riemannian foliation of M, in the Poisson case E=T^*M with a sub-Riemannian structure which is Poisson invariant with respect to its annihilator bundle. (For integrable image of S, this means that the induced Riemannian leaves should be invariant with respect to all Hamiltonian vector fields of functions which are locally constant on this foliation). This provides a huge class of new gravity models in d dimensions, embedding known 2d and 3d models as particular examples.     

Hamiltonian structure of the integrable equations under matrix Z N-reduction

The general form of the integrable equations under matrix Z _N-reduction and the corresponding infinite family of Hamiltonian structures are found.     

On the Classification of Automorphic Lie Algebras

The problem of reduction of integrable equations can be formulated in a uniform way using the theory of invariants. This provides a powerful tool of analysis and it opens the road to new applications of Automorphic Lie Algebras, beyond the context of integrable systems. In this paper it is shown that \mathfraksl₂(\mathbbC){\mathfrak{sl}_{2}(\mathbb{C})}–based Automorphic Lie Algebras associated to the icosahedral group \mathbb I{{\mathbb I}}, the octahedral group \mathbb O{{\mathbb O}}, the tetrahedral group \mathbb T{{\mathbb T}}, and the dihedral group \mathbb D_n{{\mathbb D}_n} are isomorphic. The proof is based on techniques from classical invariant theory and makes use of Clebsch-Gordan decomposition and transvectants, Molien functions and the trace-form. This result provides a complete classification of \mathfraksl₂(\mathbbC){\mathfrak{sl}_{2}(\mathbb{C})}–based Automorphic Lie Algebras associated to finite groups when the group representations are chosen to be the same and it is a crucial step towards the complete classification of Automorphic Lie Algebras.     

Binary nonlinearization of the super classical-Boussinesq hierarchy

The symmetry constraint and binary nonlinearization of Lax pairs for the super classical-Boussinesq hierarchy is obtained.Under the obtained symmetry constraint,the n-th flow of the super classical-Boussinesq hierarchy is decomposed into two super finite-dimensional integrable Hamiltonian systems,defined over the super-symmetry manifold with the corresponding dynamical variables x and t n.The integrals of motion required for Liouville integrability are explicitly given.     

A method of reduction of Einstein's equations of evolution and a natural symplectic structure on the space of gravitational degrees of freedom

A program is outlined which addresses the problem of thereduction of Einstein's equations, namely, that of writing Einstein's vacuum equations in (3+1)-dimensions as anunconstrained dynamical system where the variables are thetrue degrees of freedom of the gravitational field. Our analysis is applicable for globally hyperbolic Ricci-flat spacetimes that admit constant mean curvature compact orientable spacelike Cauchy hypersurfaces M with degM=0 andM not diffeomorphic toF ₆, the underlying manifold of a certain compact orientable flat affine 3-manifold. We find that for these spacetimes, modulo the extended Poincaré conjecture and the use of local cross-sections rather than a global cross-section, (3+1)-reduction can be completed much as in the (2+1)-dimensional case. In both cases, one gets as the reduced phase space the cotangent bundleT ^* T _M of theTeichmüller space T _M of conformal structures onM, whereM is a given initial constant mean curvature compact orientable spacelike Cauchy hypersurface in a spacetime (V, g _V ), and one gets reduction of the full classical non-reduced Hamiltonian system with constraints to a reduced Hamiltonian system without constraints onT ^* T _M . For these reduced systems, the time parameter is the parameter of a family of monotonically increasing constant mean curvature compact orientable spacelike Cauchy hypersurfaces in a neighborhood of a given initial one. In the (2+1)-dimensional case, the Hamiltonian is the area functional of these hypersurfaces, and in the (3+1)-dimensional case, the Hamiltonian is the volume functional of these hypersurfaces.     

Surfaces and the Sklyanin Bracket

 We discuss the Lie Poisson group structures associated to splittings of the loop group LGL(N,ℂ), due to Sklyanin. Concentrating on the finite dimensional leaves of the associated Poisson structure, we show that the geometry of the leaves is intimately related to a complex algebraic ruled surface with a ℂ ^*-invariant Poisson structure. In particular, Sklyanin's Lie Poisson structure admits a suitable abelianisation, once one passes to an appropriate spectral curve. The Sklyanin structure is then equivalent to one considered by Mukai, Tyurin and Bottacin on a moduli space of sheaves on the Poisson surface. The abelianization procedure gives rise to natural Darboux coordinates for these leaves, as well as separation of variables for the integrable Hamiltonian systems associated to invariant functions on the group. Received: 8 August 2001/Accepted: 29 April 2002 Published online: 14 October 2002 RID="★" ID="★" The first author of this article would like to thank NSERC and FCAR for their support RID="★★" ID="★★" The second author was partially supported by NSF grant number DMS-9802532     

Perturbed Euler top and bifurcation of limit cycles on invariant Casimir surfaces

Analytical perturbations of the Euler top are considered. The perturbations are based on the Poisson structure for such a dynamical system, in such a way that the Casimir invariants of the system remain invariant for the perturbed flow. By means of the Poincaré-Pontryagin theory, the existence of limit cycles on the invariant Casimir surfaces for the perturbed system is investigated up to first order of perturbation, providing sharp bounds for their number. Examples are given.     

Classical and quantum integrable systems in 263-1263-1263-1and separation of variables

Classical integrable Hamiltonian systems generated by elements of the Poisson commuting ring of spectral invariants on rational coadjoint orbits of the loop algebra are integrated by separation of variables in the Hamilton-Jacobi equation in hyperellipsoidal coordinates. The canonically quantized systems are then shown to also be completely integrable and separable within the same coordinates. Pairs of second class constraints defining reduced phase spaces are implemented in the quantized systems by choosing one constraint as an invariant, and interpreting the other as determining a quotient (i.e. by treating one as a first class constraint and the other as a gauge condition). Completely integrable, separable systems on spheres and ellipsoids result, but those on ellipsoids require a further modification of order in the commuting invariants in order to assure self-adjointness and to recover the Laplacian for the case of free motion. for each case — in the ambient space ⁿ , the sphere and the ellipsoid — the Schrödinger equations are completely separated in hyperellipsoidal coordinates, giving equations of generalized Lamé type.Research supported in part by the Natural Sciences and Engineering Research Council of Canada and the Fonds FCAR du Québec.