Lower-Dimensional Invariant Tori for Perturbations of a Class of Non-convex Hamiltonian Functions

We consider a class of quasi-integrable Hamiltonian systems obtained by adding to a non-convex Hamiltonian function of an integrable system a perturbation depending only on the angle variables. We focus on a resonant maximal torus of the unperturbed system, foliated into a family of lower-dimensional tori of codimension 1, invariant under a quasi-periodic flow with rotation vector satisfying some mild Diophantine condition. We show that at least one lower-dimensional torus with that rotation vector always exists also for the perturbed system. The proof is based on multiscale analysis and resummation procedures of divergent series. A crucial role is played by suitable symmetries and cancellations, ultimately due to the Hamiltonian structure of the system.     

A KAM Theorem for Hamiltonian Partial Differential Equations with Unbounded Perturbations

We establish an abstract infinite dimensional KAM theorem dealing with unbounded perturbation vector-field, which could be applied to a large class of Hamiltonian PDEs containing the derivative ? _x in the perturbation. Especially, in this range of application lie a class of derivative nonlinear Schrödinger equations with Dirichlet boundary conditions and perturbed Benjamin-Ono equation with periodic boundary conditions, so KAM tori and thus quasi-periodic solutions are obtained for them.     

Symmetry and Resonance in Periodic FPU Chains

The symmetry and resonance properties of the Fermi Pasta Ulam chain with periodic boundary conditions are exploited to construct a near-identity transformation bringing this Hamiltonian system into a particularly simple form. This “Birkhoff–Gustavson normal form” retains the symmetries of the original system and we show that in most cases this allows us to view the periodic FPU Hamiltonian as a perturbation of a nondegenerate Liouville integrable Hamiltonian. According to the KAM theorem this proves the existence of many invariant tori on which motion is quasiperiodic. Experiments confirm this qualitative behaviour. We note that one can not expect this in lower-order resonant Hamiltonian systems. So the periodic FPU chain is an exception and its special features are caused by a combination of special resonances and symmetries. Received: 25 July 2000 / Accepted: 20 December 2000     

Spectral distribution methods in effective interaction theory

Spectral distribution methods are used to suggest alternative forms for effective interaction calculation. Using the configuration centroid operator as the unperturbed Hamiltonian is found to yield a residual interaction with minimal Euclidean norm. If the effective interaction is expanded in terms of orthogonal polynomials appropriate to the configuration density, the leading term is more meaningful than those in existing series. Using the same technique of orthogonal polynomial expansion, the calculation of diagonal matrix elements of higher order terms in the perturbation expansions reduces to an evaluation of traces which are more easily calculable. Convergence of both polynomial expansions is tied to the convergence of the spectral density to normal form which itself is governed by a strong principle, the central limit theorem.     

Degenerate Elliptic Resonances

Quasi-periodic motions on invariant tori of an integrable system of dimension smaller than half the phase space dimension may continue to exist after small perturbations. The parametric equations of the invariant tori can often be computed as a formal power series in the perturbation parameter and can be given a meaning via resummations. Here we prove that, for a class of elliptic tori, a resummation algorithm can be devised and proved to be convergent, thus extending to such lower-dimensional invariant tori the methods employed to prove convergence of the Lindstedt series either for the maximal (i.e. KAM) tori or for the hyperbolic lower-dimensional invariant tori.     

Convergence properties of the Brillouin-Wigner type of perturbation expansion

We study the Brillouin-Wigner perturbation expansion of the model-space effective Hamiltonian corresponding to the full Hamiltonian H(x) = H₀ + xH₁, H₀ and H₁ being respectively the unperturbed and the interaction Hamiltonian and x being a strength parameter. The radius of convergence for the perturbation expansion is related to the poles of the energy-dependent effective interaction, and the location of these poles in the complex x-plane is discussed. The situation with poles lying off the real x-axis is examined. In terms of the spectrum of the unperturbed Hamiltonian H₀, some necessary conditions for convergence are derived, and the effects of intruder states are discussed. It is shown that the BW expansion of the ground-state energy can always be made convergent by a shift of the unperturbed energy spectrum.     

Nonequilibrium dynamics of strings in time-dependent plane wave backgrounds

We formulate and study the nonequilibrium dynamics of strings near the singularity of the time-dependent plane wave background in the framework of the Nonequilibrium Thermo Field Dynamics (NETFD). In particular, we construct the Hilbert space of the thermal string oscillators at nonequilibrium and generalize the NETFD to describe the coordinates of the center of mass of the thermal string. The equations of motion of the thermal fields and the Hamiltonian are derived. Due to the time-dependence of the oscillator frequencies, a counterterm is present in the Hamiltonian. This counterterm determines the correlation functions in a perturbative fashion. We compute the two point correlation function of the thermal string at zero order in the power expansion.     

Ray dynamics in long-range deep ocean sound propagation

Recent results relating to ray dynamics in ocean acoustics are reviewed. Attention is focused on long-range propagation in deep ocean environments. For this class of problems, the ray equations may be simplified by making use of a one-way formulation in which the range variable appears as the independent (timelike) variable. Topics discussed include integrable and nonintegrable ray systems, action-angle variables, nonlinear resonances and the KAM theorem, ray chaos, Lyapunov exponents, predictability, nondegeneracy violation, ray intensity statistics, semiclassical breakdown, wave chaos, and the connection between ray chaos and mode coupling. The Hamiltonian structure of the ray equations plays an important role in all of these topics.     

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     

Localization in the Ground State of an Interacting Quasi-Periodic Fermionic Chain

We consider a one dimensional many body fermionic system with a large incommensurate external potential and a weak short range interaction. We prove, for chemical potentials in a gap of the non interacting spectrum, that the zero temperature thermodynamical correlations are exponentially decaying for large distances, with a decay rate much larger than the gap; this indicates the persistence of localization in the interacting ground state. The analysis is based on the renormalization group, and convergence of the renormalized expansion is achieved using fermionic cancellations and controlling the small divisor problem assuming a Diophantine condition for the frequency.     

Stability for Quasi-Periodically Perturbed Hill's Equations

We consider a perturbed Hill's equation of the form +(p₀(t)+ɛp₁(t))ϕ=0, where p₀ is real analytic and periodic, p₁ is real analytic and quasi-periodic and ɛ ∈ℝ is ``small'. Assuming Diophantine conditions on the frequencies of the decoupled system, i.e. the frequencies of the external potentials p₀ and p₁ and the proper frequency of the unperturbed (ɛ=0) Hill's equation, but without making any assumptions on the perturbing potential p₁ other than analyticity, we prove that quasi-periodic solutions of the unperturbed equation can be continued into quasi-periodic solutions if ɛ lies in a Cantor set of relatively large measure in where ɛ₀ is small enough. Our method is based on a resummation procedure of a formal Lindstedt series obtained as a solution of a generalized Riccati equation associated to Hill's problem.     

On the problem of stability for higher-order derivative Lagrangian systems

The problem of stability for dynamical systems whose Lagrangian function depends on the derivatives of a higher order than one is studied. The difficulty of this analysis arises from the indefiniteness of the Hamiltonian, so that the well-known Lagrange-Dirichlet theorem cannot be used and the methods of the canonical perturbation theory (KAM theory) must be employed. We show, with an example, that the indefiniteness of the energy does not forbid the stability.     

A KAM Theorem for Hyperbolic-Type Degenerate Lower Dimensional Tori in Hamiltonian Systems

A KAM theorem for degenerate lower dimensional tori in nearly integrable Hamiltonian systems is given in this paper. For the non-degenerate cases, both hyperbolic and elliptic, the KAM theorem has been well established by many authors ([8, 9, 11, 13, 14, 17]). Received: 23 October 1996 / Accepted: 24 June 1997     

Estimates of stability time for nearly integrable systems with a quasiconvex Hamiltonian

A Hamiltonian system differing from an integrable system by a small perturbation equals, similar varepsilon is analyzed. According to the Nekhoroshev theorem, the changes in the perturbed motion of the "action" variables of the unperturbed system are small over a time interval which increases exponentially in length as varepsilon decreases linearly. If the unperturbed Hamiltonian is a quasiconvex function of these "actions," the changes in them remain small ( equals, similar varepsilon (1/2n)) over a time interval on the order of exp(const/ varepsilon (1/2n)), where n is the number of degrees of freedom of the system.     

Weakly Mixing Invariant Tori of Hamiltonian Systems

We note that every finite or infinite dimensional real-analytic Hamiltonian system with a quasi-periodic invariant KAM torus of finite dimension d≥ 2 can be perturbed in such a way that the new real-analytic Hamiltonian system has a weakly mixing invariant torus of the same dimension. Received: 24 April 1998/ Accepted: 14 January 1999     

Compensations in small divisor problems

A general direct method, alternative to KAM theory, apt to deal with small divisor problems in the real-analytic category, is presented and tested on several small divisor problems including the construction of maximal quasi-periodic solutions for nearly-integrable non-degenerate Hamiltonian or Lagrangian systems and the construction of lower dimensional resonant tori for nearly-integrable Hamiltonian systems. The method is based on an explicit graph theoretical representation of the formal power series solutions, which allows to prove compensations among the monomials forming such representation.L.C. thanks C. Simó and theCentre de Recerca Matemàtica (Bellaterra) for kind hospitality; he also acknowledges partial support by CNR-GNAFA. The authors gratefully acknowledge helpful discussions with C. Liverani.     

Pure Point Spectrum for Two-Level Systems in a Strong Quasi-Periodic Field

We consider two-level atoms in a strong external quasi-periodic field with Diophantine frequency vector. We show that if the field is an analytic function with zero average, then for a large set of values of its frequency vector, characterized by imposing infinitely many Diophantine conditions, the spectrum of the quasi-energy operator is pure point, as in the case of nonzero average which was already known in literature.     

Optimal oscillation-center transformations

Systems with Hamiltonians of the form H₀(p) + H₁(q,p,t) are considered. A variational principle is proposed for defining that canonical transformation, continuously connected with the identity transformation, which minimizes the residual, coordinate-dependent part of the new Hamiltonian. The principle is based on minimization of the mean square generalized force over phase space and time. The transformation reduces to the action-angle transformation in that part of the phase space of an integrable system where the orbit topology is that of the unperturbed system, or on primary KAM surfaces. General arguments in favour of this definition are given, based on Galilean invariance, decay of the Fourier spectrum, and its ability to include external fields or inhomogeneous systems. The optimal oscillation-center transformation for the physical pendulum (or particle in a sinusoidal potential) is constructed analytically. A modified principle for relativistic systems is presented in an appendix.     

The Intrinsic Stochasticity of Near-Integrable Hamiltonian Systems

Under certain conditions, the dynamics of near-integrable Hamiltonian systems appears to be stochastic. This stochasticity (intrinsic stochasticity, or deterministic chaos) is closely related to the Kolmogorov-Arnold-Moser (KAM) theorem of the stability of near-integrable multiperiodic Hamiltonian systems. The effect of the intrinsic stochasticity attracts still growing attention both in theory and in various applications in contemporary physics. The paper discusses the relation of the intrinsic stochasticity to the modern ergodic theory and to the KAM theorem, and describes some numerical experiments on related astrophysical and high-temperature plasma problems. Some open questions are mentioned in Conclusion.     

Symmetries in constrained Hamiltonian systems

We derive an algorithm for the construction of all the gauge generators of a constrained hamiltonian theory. Dirac's conjecture that all secondary first-class constraints generate symmetries is revisited and replaced by a theorem. The algorithm is applied to Yang-Mills theories and metric gravity, and we find generators which operate on the complete set of canonical variables, thus producing the correct transformation laws also for the unphysical coordinates. Finally we discuss the general structure of the Hamiltonian for constrained theories. We show how in most cases one can read off the first-class constraints directly from the Hamiltonian.