Energy Transfer in a Fast-Slow Hamiltonian System

We consider a finite region of a lattice of weakly interacting geodesic flows on manifolds of negative curvature and we show that, when rescaling the interactions and the time appropriately, the energies of the flows evolve according to a nonlinear diffusion equation. This is a first step toward the derivation of macroscopic equations from a Hamiltonian microscopic dynamics in the case of weakly coupled systems.     

Black holes as generalised Toda molecules

In this note we compare the geodesic formalism for spherically symmetric black hole solutions with the black hole effective potential approach. The geodesic formalism is beneficial for symmetric supergravity theories since the symmetries of the larger target space lead to a complete set of commuting constants of motion that establish the integrability of the geodesic equations of motion, as shown in arXiv:1007.3209. We point out that the integrability lifts straightforwardly to the integrability of the equations of motion with a black hole potential. This construction turns out to be a generalisation of the connection between Toda molecule equations and geodesic motion on symmetric spaces known in the mathematics literature. We describe in some detail how this generalisation of the Toda molecule equations arises.     

Regular and irregular geodesics on spherical harmonic surfaces

The behavior of geodesic curves on even seemingly simple surfaces can be surprisingly complex. In this paper we use the Hamiltonian formulation of the geodesic equations to analyze their integrability properties. In particular, we examine the behavior of geodesics on surfaces defined by the spherical harmonics. Using the Morales-Ramis theorem and Kovacic algorithm we are able to prove that the geodesic equations on all surfaces defined by the sectoral harmonics are not integrable, and we use Poincaré sections to demonstrate the breakdown of regular motion.     

r-Matrix Structure for a Restricted Flow with Bargmann Constraint

This paper deals with the integrability of a finite-dimensional Hamiltonian system linked with the generalized coupled KdV hierarchy. For this purpose the associated Lax representation is presented after an elementary calculation. It is shown that the Lax representation enjoys a dynamical r-matrix formula instead of a classical one in the Poisson bracket on R^2N. Consequently the resulting system is proved to be completely integrable in view of its r-matrix structure.     

r-Matrix Structure for a Restricted Flow with Bargmann Constraint

This paper deals with the integrability of a finite-dimensional Hamiltonian system linked with the generalized coupled KdV hierarchy. For this purpose the associated Lax representation is presented after an elementary calculation. It is shown that the Lax representation enjoys a dynamical r-matrix formula instead of a classical one in the Poisson bracket on R2N. Consequently the resulting system is proved to be completely integrable in view of its r-matrix structure.     

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     

Isospectral deformations of random Jacobi operators

Summary We show the integrability of infinite dimensional Hamiltonian systems obtained by making isospectral deformations of random Jacobi operators over an abstract dynamical system. The time 1 map of these so called random Toda flows can be expressed by aQR decomposition.     

Unbounded Energy Growth in Hamiltonian Systems with a Slowly Varying Parameter

We study Hamiltonian systems which depend slowly on time. We show that if the corresponding frozen system has a uniformly hyperbolic invariant set with chaotic behaviour, then the full system has orbits with unbounded energy growth (under very mild genericity assumptions). We also provide formulas for the calculation of the rate of the fastest energy growth. We apply our general theory to non-autonomous perturbations of geodesic flows and Hamiltonian systems with billiard-like and homogeneous potentials. In these examples, we show the existence of orbits with the rates of energy growth that range, depending on the type of perturbation, from linear to exponential in time. Our theory also applies to non-Hamiltonian systems with a first integral.     

On the Stability of Double Homoclinic Loops

We consider 2-degrees of freedom Hamiltonian systems with an involutive symmetry and a pair of orbits bi-asymptotic (homoclinic) to a saddle-center equilibrium (related to pairs of pure real, ±ν, and pure imaginary eigenvalues, ±ω i). We show that the stability of this double homoclinic loop is determined by the reflection coefficient of a one-dimensional scattering problem and ω/ν. We also show that the mechanism for losing stability is the creation of an infinite heteroclinic chain connecting a sequence of periodic orbits that accumulates at the double loop. Received: 10 November 1995 / Accepted: 5 June 1996     

Reductions to Korteweg-de Vries Soliton Hierarchy

Based on the nonlinearization of Lax pairs, the Korteweg-de Vries (KdV) soliton hierarchy is decomposed into a family of finite-dimensional Hamiltonian systems, whose Liouville integrability is proved by means of the elliptic coordinates. By applying the Abel-Jacobi coordinates on a Riemann surface of hyperelliptic curve, the resulting Hamiltonian flows as well as the KdV soliton hierarchy are ultimately reduced into linear superpositions, expressed by the Abel-Jacobi variables.     

Hamiltonian operators and ℓ-coverings

An efficient method to construct Hamiltonian structures for nonlinear evolution equations is described. It is based on the notions of variational Schouten bracket and ℓ^*-covering. The latter serves the role of the cotangent bundle in the category of nonlinear evolution PDEs. We first consider two illustrative examples (the KdV equation and the Boussinesq system) and reconstruct for them the known Hamiltonian structures by our methods. For the coupled KdV–mKdV system, a new Hamiltonian structure is found and its uniqueness (in the class of polynomial (x,t)-independent structures) is proved. We also construct a nonlocal Hamiltonian structure for this system and prove its compatibility with the local one.     

On the Riemannian description of chaotic instability in Hamiltonian dynamics

In this work we investigate Hamiltonian chaos using elementary Riemannian geometry. This is possible because the trajectories of a standard Hamiltonian system (i.e., having a quadratic kinetic energy term) can be seen as geodesics of the configuration space manifold equipped with the standard Jacobi metric. The stability of the dynamics is tackled with the Jacobi-Levi-Civita equation (JLCE) for geodesic spread and is applied to the case of a two degrees of freedom Hamiltonian. A detailed comparison is made among the qualitative informations given by Poincare sections and the results of the geometric investigation. Complete agreement is found. The solutions of the JLCE are also in quantitative agreement with the solutions of the tangent dynamics equation. The configuration space manifold associated to the Hamiltonian studied here is everywhere of positive curvature. However, curvature is not constant and its fluctuations along the geodesics can yield parametric instability of the trajectories, thus chaos. This mechanism seems to be one of the most effective sources of chaotic instabilities in Hamiltonians of physical interest, and makes a major difference with Anosov flows, and, in general, with abstract geodesic flows of ergodic theory. (c) 1995 American Institute of Physics.     

Hamiltonian and gradient structures in the Toda flows

In this paper we consider gradient structures in the dynamics and geometry of the asymmetri nonperiodic tridiagonal and full Toda flow equations. We compare and contrast a number of formulations of the nonperiodic Toda equations. In the case of the full Kostant (asymmetric) Toda flow we explain the role of noncommutative integrability in its qualitative behavior. We describe the relationship between the asymmetric Toda flows and the symmetric and indefinite Toda flows, and prove in particular that one may conjugate from the full Kostant Toda flows to the full symmetric Toda flows via a Poisson map.     

Classical Mechanics in Hilbert Space,Part 1

We consider the Hamilton formulation as well as the Hamiltonian flows on a symplectic (phase) space. These symplectic spaces are derivable from the Lie group of symmetries of the physical system considered. In Part 2 of this work, we then obtain the Hamiltonian formalism in the Hilbert spaces of square integrable functions on the symplectic spaces so obtained.     

A Suggestion for an Integrability Notion for Two-Dimensional Spin Systems

We suggest that trialgebraic symmetries might be a sensible starting point for a notion of integrability for two dimensional spin systems. For a simple trialgebraic symmetry we give an explicit condition in terms of matrices which a Hamiltonian realizing such a symmetry has to satisfy and give an example of such a Hamiltonian which realizes a trialgebra recently given by the authors in another paper. Besides this, we also show that the same trialgebra can be realized on a kind of Fock space of q-oscillators, i.e. the suggested integrability concept gets via this symmetry a close connection to a kind of noncommutative quantum field theory, paralleling the relation between the integrability of spin chains and two dimensional conformal field theory.     

A Spin-Shift Operator of the Multi-particle Ruijsenaars–Schneider Hamiltonian

We obtain a spin-shift operator for the multi-particle trigonometric Ruijsenaars–Schneider Hamiltonian. This result is a generalization of the argument in Phys. Lett. B 375 (1996), 89–97, where the integrability of the one-particle Ruijsenaars–Schneider system is shown by using the existence of a spin-shift operator.     

Hamiltonian systems on fibered manifolds

We offer a new geometric theory of Hamiltonian systems with an infinite number of degrees of freedom in which the Hamiltonian operators are nonlinear differential operators on fields. The Poisson bracket is carried into the vertical bracket by the mapping between functionals and Hamitonian operators which is established by a Hamiltonian structure.