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.     

KAM Theory in Configuration Space and Cancellations in the Lindstedt Series

The KAM theorem for analytic quasi-integrable anisochronous Hamiltonian systems yields that the perturbation expansion (Lindstedt series) for any quasi-periodic solution with Diophantine frequency vector converges. If one studies the Lindstedt series by following a perturbation theory approach, one finds that convergence is ultimately related to the presence of cancellations between contributions of the same perturbation order. In turn, this is due to symmetries in the problem. Such symmetries are easily visualised in action-angle coordinates, where the KAM theorem is usually formulated by exploiting the analogy between Lindstedt series and perturbation expansions in quantum field theory and, in particular, the possibility of expressing the solutions in terms of tree graphs, which are the analogue of Feynman diagrams. If the unperturbed system is isochronous, Moser’s modifying terms theorem ensures that an analytic quasi-periodic solution with the same Diophantine frequency vector as the unperturbed Hamiltonian exists for the system obtained by adding a suitable constant (counterterm) to the vector field. Also in this case, one can follow the alternative approach of studying the perturbation expansion for both the solution and the counterterm, and again convergence of the two series is obtained as a consequence of deep cancellations between contributions of the same order. In this paper, we revisit Moser’s theorem, by studying the perturbation expansion one obtains by working in Cartesian coordinates. We investigate the symmetries giving rise to the cancellations which makes possible the convergence of the series. We find that the cancellation mechanism works in a completely different way in Cartesian coordinates, and the interpretation of the underlying symmetries in terms of tree graphs is much more subtle than in the case of action-angle coordinates.     

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     

A KAM Theorem for Hamiltonian Partial Differential Equations in Higher Dimensional Spaces

In this paper, we give a KAM theorem for a class of infinite dimensional nearly integrable Hamiltonian systems. The theorem can be applied to some Hamiltonian partial differential equations in higher dimensional spaces with periodic boundary conditions to construct linearly stable quasi–periodic solutions and its local Birkhoff normal form. The applications to the higher dimensional beam equations and the higher dimensional Schrödinger equations with nonlocal smooth nonlinearity are also given in this paper.     

Numerical study of a billiard in a gravitational field

Billiards have always been used as models for mechanical systems. In this paper we describe a very simple billiard which, over a range of one continous parameter only, exhibits the characteristics of Hamiltonian systems having two degrees of freedom and a discontinuity. The relationship between this billiard and the well-known one-dimensional self-gravitating system (with N = 3) is given. This billiard consists of a mass point moving in a symmetric wedge of angle 2θ under the influence of a constant gravitational field. For θ<45° KAM and chaotic regions coexist in the phase space. A specific family of curves, related to collisions at the wedge vertex, limits the expansion of near-integrable regions. For θ=45°, the motion is strictly integrable. Finally, for θ>;45°, complete chaos is obtained, suggesting K-system behavior. The general properties of the mapping and some numerical results obtained are discussed. Of special interest are invariant curves which cross a line of discontinuity, and a new “universality” class for Lyapunov numbers.     

Renormalization method for computing the threshold of the large-scale stochastic instability in two degrees of freedom Hamiltonian systems

An approximate renormalization procedure is derived for the HamiltonianH(v,x,t)=v²/2–M cosx–P cosk(x–t). It gives an estimate of the large scale stochastic instability threshold which agrees within 5–10% with the results obtained from direct numerical integration of the canonical equations. It shows that this instability is related to the destruction of KAM tori between the two resonances and makes the connection with KAM theory. Possible improvements of the method are proposed. The results obtained forH allow us to estimate the threshold for a large class of Hamiltonian systems with two degrees of freedom.     

A λ-Lemma for Partially Hyperbolic Tori and the Obstruction Property

Using a scheme given by Marco, we prove that partially hyperbolic tori along resonant surfaces of near-integrable Hamiltonian systems possess the obstruction property in Arnold's terminology. The proof is based on a specific lambda lemma for these tori.     

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     

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.     

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.     

Branching of Cantor Manifolds of Elliptic Tori and Applications to PDEs

We consider infinite dimensional Hamiltonian systems. We prove the existence of “Cantor manifolds” of elliptic tori–of any finite higher dimension–accumulating on a given elliptic KAM torus. Then, close to an elliptic equilibrium, we show the existence of Cantor manifolds of elliptic tori which are “branching” points of other Cantor manifolds of higher dimensional tori. We also answer to a conjecture of Bourgain, proving the existence of invariant elliptic tori with tangential frequency along a pre-assigned direction. The proofs are based on an improved KAM theorem. Its main advantages are an explicit characterization of the Cantor set of parameters and weaker smallness conditions on the perturbation. We apply these results to the nonlinear wave equation.     

Undecidability and incompleteness in classical mechanics

We describe Richardson's functor from the Diophantine equations and Diophantine problems into elementary real-valued functions and problems. We then derive a general undecidability and incompleteness result for elementary functions within ZFC set theory, and apply it to some problems in Hamiltonian mechanics and dynamical systems theory. Our examples deal with the algorithmic impossibility of deciding whether a given Hamiltonian can be integrated by quadratures and related questions; they lead to a version of Gödel's incompleteness theorem within Hamiltonian mechanics. A similar application to the unsolvability of the decision problem for chaotic dynamical systems is also obtained.     

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.     

The KAM theory of systems with short range interactions,II

The proof of the results on the KAM theory of systems with short range interactions, stated in [6] is completed. Estimates on the decay of the interactions generated by the iterative procedure in the KAM theorem are proved, as well as the modification of the theorems of [2–3] needed for results.This work was completed while the author was at the Institute for Mathematics and Its Applications, University of Minnesota, Minneapolis, MN, 55455, USASupported in part by NSF Grant DMS-8403664     

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.     

An Approximate KAM-Renormalization-Group Scheme for Hamiltonian Systems

We construct an approximate renormalization scheme for Hamiltonian systems with two degrees of freedom. This scheme is a combination of Kolmogorov–Arnold–Moser (KAM) theory and renormalization-group techniques. It makes the connection between the approximate renormalization procedure derived by Escande and Doveil and a systematic expansion of the transformation. In particular, we show that the two main approximations, consisting in keeping only the quadratic terms in the actions and the two main resonances, keep the essential information on the threshold of the breakup of invariant tori.     

统计物理的微观动力学基础

统计的基本出发点是研究系统具有的随机性,不同系统在不同情形下的宏观热力学性质起源于系统内部随机性的差异,通过对宏观热力学系统的微观非线性动力学进行研究探索，我们可以进一步更为深入地理解物态方程、相变等诸多的宏观热力学现象。本文通过哈密顿系统的非线性动力学研究，以及遍历性理论的动力学随机性研究对此问题进行了分析，研究表明，动力学系统的全局性混沌是系统统计成立的根本要素，系统的无限大自由度（热力学极限）已不是决定性的因素，人们可以在此基础上建立少自由度系统的统计力学及热力学。     

Controlling global stochasticity of Hamiltonian systems by nonlinear perturbation

A method of controlling global stochasticity in Hamiltonian systems by applying nonlinear perturbation is proposed. With the well-known standard map we demonstrate that this control method can convert global stochasticity into regular motion in a wide chaotic region for arbitrary initial condition, in which the control signal remains very weak after a few kicks. The system in which chaos has been controlled approximates to the original Hamiltonian system, and this approach appears robust against small external noise. The mechanism underlying this high control efficiency is intuitively explained. Received 15 January 2002 Published online 6 June 2002     

Birkhoff-Kolmogorov-Arnold-Moser tori in convex Hamiltonian systems

For the Hamiltonian systems of KAM type, it is proved that some lower dimensional invariant tori always exist in the resonance gaps although those maximum tori can not survive small perturbations in the generic case.     

Absence of diffusion in the Anderson tight binding model for large disorder or low energy

We prove that the Green's function of the Anderson tight binding Hamiltonian decays exponentially fast at long distances on ? ^v , with probability 1. We must assume that either the disorder is large or the energy is sufficiently low. Our proof is based on perturbation theory about an infinite sequence of block Hamiltonians and is related to KAM methods.