Birkhoffian symplectic algorithms derived from Hamiltonian symplectic algorithms

In this paper, we focus on the construction of structure preserving algorithms for Birkhoffian systems, based on existing symplectic schemes for the Hamiltonian equations. The key of the method is to seek an invertible transformation which drives the Birkhoffian equations reduce to the Hamiltonian equations. When there exists such a transformation,applying the corresponding inverse map to symplectic discretization of the Hamiltonian equations, then resulting difference schemes are verified to be Birkhoffian symplectic for the original Birkhoffian equations. To illustrate the operation process of the method, we construct several desirable algorithms for the linear damped oscillator and the single pendulum with linear dissipation respectively. All of them exhibit excellent numerical behavior, especially in preserving conserved quantities.     

Exponential stability of states close to resonance in infinite-dimensional Hamiltonian systems

We develop canonical perturbation theory for a physically interesting class of infinite-dimensional systems. We prove stability up to exponentially large times for dynamical situations characterized by a finite number of frequencies. An application to two model problems is also made. For an arbitrarily large FPU-like system with alternate light and heavy masses we prove that the exchange of energy between the optical and the acoustical modes is frozen up to exponentially large times, provided the total energy is small enough. For an infinite chain of weakly coupled rotators we prove exponential stability for two kinds of initial data: (a) states with a finite number of excited rotators, and (b) states with the left part of the chain uniformly excited and the right part at rest.     

Stability of motions near resonances in quasi-integrable Hamiltonian systems

Nekhoroshev's theorem on the stability of motions in quasi-integrable Hamiltonian systems is revisited. At variance with the proofs already available in the literature, we explicitly consider the case of weakly perturbed harmonic oscillators; furthermore we prove the confinement of orbits in resonant regions, in the general case of nonisochronous systems, by using the elementary idea of energy conservation instead of more complicated mechanisms. An application of Nekhoroshev's theorem to the study of perturbed motions inside resonances is also provided.Partially supported by Ministere della Pubblica Istruzione.Partially supported by Grant N.S.F. DMS 85-03333 and by Ministero della Pubblica Istruzione.     

Symplectic integration of nonlinear Hamiltonian systems

There exist several standard numerical methods for integrating ordinary differential equations. However, if one is interested in integration of Hamiltonian systems, these methods can lead to wrong results. This is due to the fact that these methods do not explicitly preserve the so-called ‘symplectic condition’ (that needs to be satisfied for Hamiltonian systems) at every integration step. In this paper, we look at various methods for integration that preserve the symplectic condition.     

The symplectic eigenfunction expansion theorem and its application to the plate bending equation

This paper deals with the bending problem of rectangular plates with two opposite edges simply supported. It is proved that there exists no normed symplectic orthogonal eigenfunction system for the associated infinite-dimensional Hamiltonian operator H and that the two block operators belonging to Hamiltonian operator H possess two normed symplectic orthogonal eigenfunction systems in some space. It is demonstrated by using the properties of the block operators that the above bending problem can be solved by the symplectic eigenfunction expansion theorem, thereby obtaining analytical solutions of rectangular plates with two opposite edges simply supported and the other two edges supported in any manner.     

An Averaging Theorem for Hamiltonian Dynamical Systems in the Thermodynamic Limit

It is shown how to perform some steps of perturbation theory if one assumes a measure-theoretic point of view, i.e. if one renounces to control the evolution of the single trajectories, and the attention is restricted to controlling the evolution of the measure of some meaningful subsets of phase–space. For a system of coupled rotators, estimates uniform in N for finite specific energy can be obtained in quite a direct way. This is achieved by making reference not to the sup norm, but rather, following Koopman and von Neumann, to the much weaker L ² norm.     

Estimates for Non-Resonant Normal Forms in Hamiltonian Perturbation Theory

We make a remark about an estimate of the rest for the non-resonant action-angle normal forms and exhibit a simple example suggesting the optimality of this estimate when there are no small divisors. Given a polynomial perturbation of degree P and an integer k, calling the size of the small denominators up to order k, we prove that the kth order remainder is bounded by (2/ ₀) ^k+1 with ₀=const ²/(kP ²). Thus, fixing the degree of the perturbation, if is independent of k (i.e., if there are no small divisors), we obtain a rest bounded by (const k) ^k+1. These estimates are also applied to the case in which the small divisors are absent, and they are conjectured to be optimal in this context. To support this idea we present a simplified model problem with no small denominators, formally related to the above calculations, and we show that it indeed has factorial divergence of its Birkhoff series. We also obtain Nekhoroshev's Theorem for harmonic oscillators. We hope that our simple approach makes more accessible to a general audience this important (although quite technical) topic.     

On the spectra of periodic waves for infinite-dimensional Hamiltonian systems

We consider the problem of determining the spectrum for the linearization of an infinite-dimensional Hamiltonian system about a spatially periodic traveling wave. By using a Bloch-wave decomposition, we recast the problem as determining the point spectra for a family of operators J_γL_γ, where J_γ is skew-symmetric with bounded inverse and L_γ is symmetric with compact inverse. Our main result relates the number of unstable eigenvalues of the operator J_γL_γ to the number of negative eigenvalues of the symmetric operator L_γ. The compactness of the resolvent operators allows us to greatly simplify the proofs, as compared to those where similar results are obtained for linearizations about localized waves. The theoretical results are general, and apply to a larger class of problems than those considered herein. The theory is applied to a study of the spectra associated with periodic and quasi-periodic solutions to the nonlinear Schrödinger equation, as well as periodic solutions to the generalized Korteweg-de Vries equation with power nonlinearity.     

完全可积的非线性方程建立哈密顿理论的一般方法和对SG方程应用

完全可积的非线性方程的单式矩阵的泊松括号已知可以表为对x的积分，指出被积函数一定 可以表为约斯特解对的直积的线性组合的微分，并可由直积矩阵相应元的对比确定组合系数 .从而解决了建立非线性方程哈密顿理论的一般方法.由于实验室系中的SG方程，相应的表述 异常复杂，所以以它为例来说明方法的实质.同时由于现有的相关工作违反了泊松括号同时 性的要求，给出了必要的改正. 关键词： 非线性方程 哈密顿理论 孤子     

On the spectrum of the Heisenberg Hamiltonian

The quantum, antiferromagnetic, spin-1/2 Heisenberg Hamiltonian on thed-dimensional cubic lattice ^d is considered for any dimensiond. First the anisotropic case is considered for small transversal coupling and a convergent expansion is given for a family of eigenprojections which is complete in all finite-volume truncations. Then the general case is considered, for which an upper bound to the ground-state energy is given which is optimal for strong enough anisotropy. This bound is expressed through a functional involving the statistical expectation value at finite temperature of a certain correlation function of an Ising model defined on the lattice ^d itself.     

对声波和弹性波传播模拟的Hamilton系统方法

建立了离散化网格上的准粒子体系，引入此体系的Hamilton系统描述，用来模拟声波和弹性波的传播.介绍了准粒子间相互作用的九点模型并给出互作用系数.证明了Hamilton系统方法 和声波、弹性波方程的关系，并给出两个方法中所使用物理量的关系.使用辛算法对给定的 介质模型进行数值模拟. 关键词： Hamilton系统方法 九点互作用模型 声波方程 弹性波方程 辛算法     

Time dependent canonical perturbation theory III: Application to a system with nonconstant unperturbed frequencies

In this communication, we report the results of the application of time dependent perturbation theory to a non-integrable Hamiltonian which is a perturbation on a Hamiltonian with nonconstant frequencies. The theory provides good time dependent local constants of motion and also gives good approximation for mapping of solutions for a time limit determined by the nearest singularity in complexε plane for fixed real time and the order of calculation.     

Large Deviations of Lattice Hamiltonian Dynamics Coupled to Stochastic Thermostats

We discuss the Donsker-Varadhan theory of large deviations in the framework of Hamiltonian systems thermostated by a Gaussian stochastic coupling. We derive a general formula for the Donsker-Varadhan large deviation functional for dynamics which satisfy natural properties under time reversal. Next, we discuss the characterization of the stationary states as the solution of a variational principle and its relation to the minimum entropy production principle. Finally, we compute the large deviation functional of the current in the case of a harmonic chain thermostated by a Gaussian stochastic coupling.     

The quadratic-form identity for constructing Hamiltonian structures of the Guo hierarchy

The trace identity is extended to the quadratic-form identity. The Hamiltonian structures of the multi-component Guo hierarchy, integrable coupling of Guo hierarchy and (2+1)-dimensional Guo hierarchy are obtained by the quadratic-form identity. The method can be used to produce the Hamiltonian structures of the other integrable couplings or multi-component hierarchies.     

Weakly nonlinear dynamics in noncanonical Hamiltonian systems with applications to fluids and plasmas

A method, called beatification, is presented for rapidly extracting weakly nonlinear Hamiltonian systems that describe the dynamics near equilibria of systems possessing Hamiltonian form in terms of noncanonical Poisson brackets. The procedure applies to systems like fluids and plasmas in terms of Eulerian variables that have such noncanonical Poisson brackets, i.e., brackets with nonstandard and possibly degenerate form. A collection of examples of both finite and infinite dimensions is presented.     

The Principle of Covariance and the Hamiltonian Formulation of General Relativity

The implications of the general covariance principle for the establishment of a Hamiltonian variational formulation of classical General Relativity are addressed. The analysis is performed in the framework of the Einstein-Hilbert variational theory. Preliminarily, customary Lagrangian variational principles are reviewed, pointing out the existence of a novel variational formulation in which the class of variations remains unconstrained. As a second step, the conditions of validity of the non-manifestly covariant ADM variational theory are questioned. The main result concerns the proof of its intrinsic non-Hamiltonian character and the failure of this approach in providing a symplectic structure of space-time. In contrast, it is demonstrated that a solution reconciling the physical requirements of covariance and manifest covariance of variational theory with the existence of a classical Hamiltonian structure for the gravitational field can be reached in the framework of synchronous variational principles. Both path-integral and volume-integral realizations of the Hamilton variational principle are explicitly determined and the corresponding physical interpretations are pointed out.     

Hamiltonian Formalism of the Derivative Nonlinear Schrodinger Equation

A particular form of poisson bracket is introduced for the derivative nonlinear Schrodinger (DNLS) equation.And its Hamiltonian formalism is developed by a linear combination method. Action-angle variables are found.     

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.