特殊二阶微分方程的线性自治系统的辛多步法

１．引言１９世纪Ｈａｍｉｌｔｏｎ提出经典力学的一种基本的方程形式．记Ｐ＝Ｍｑ表示动量,ｑ表示位移,于是动能Ｔ＝１／２（ｐ,Ｍ－１　ｐ）,并将总能量Ｈ＝Ｔ＋Ｖ位能）表示为ｐ,ｑ的函数：于是经典力学的标准形式－Ｎｅｗｔｏｎ形式称为Ｈａｍｉｌｔｏｎ正则方程．（１．１）是ＺＩ维相空间或称辛空间中的相变量（ＰＩ,...,Ｐ。,ｑｌ;...,ｑｎ）'的一阶微分方程组．其中ｑ-（ｑｌ,...;ｑｒｔ）'是位置向量,ｄｑ,、Ｉ。。。。。ｑ＝ｉ＝（ｑｌ,...,ｑｈ）'是速度向量,ｄｔ'"""。'。'。。。ｌ、。。ｄ"ｑ,...．、,-,-…     

Construction of Symmetric and Symplectic Bilinear Forms

Let V be a finite-dimensional vector-space. A linear mapping on V is called simple if V( - 1) is 1-dimensional. Let S be a set of simple bijections on V. We discuss conditions entraining that each element of S is orthogonal (respectively symplectic) under an appropriate symmetric (respectively symplectic) bilinear form on V.     

Energy Conservation with Non-Symplectic Methods: Examples and Counter-Examples

Energy conservation of numerical integrators is well understood for symplectic one-step methods. This article provides new insight into energy conservation with non-symplectic methods. Sufficient conditions and counter-examples are presented. AMS subject classification (2000) 65L06, 65P10, 37J99.Submitted June 2004. Accepted October 2004. Communicated by Syvert Nørsett.     

Unconventional Hamilton-type variational principle in phase space and symplectic algorithm

By a novel approach proposed by Luo, the unconventional Hamilton-type variational principle in phase space for elastodynamics of multidegree-of-freedom system is established in this paper. It not only can fully characterize the initial-value problem of this dynamic, but also has a natural symplectic structure. Based on this variational principle, a symplectic algorithm which is called a symplectic time-subdomain method is proposed. A non-difference scheme is constructed by applying Lagrange interpolation polynomial to the time subdomain. Furthermore, it is also proved that the presented symplectic algorithm is an unconditionally stable one. From the results of the two numerical examples of different types, it can be seen that the accuracy and the computational efficiency of the new method excel obviously those of widely used Wilson-θ and Newmark-β methods. Therefore, this new algorithm is a highly efficient one with better computational performance.     

On generalized hamiltonian systems

1. PreliminaryIt is well known that{1] a 8ymPlectic form is invariant along the trajectory of a Hamilto-nian system. Based on this fundamental property, certain techniques have been developed.The purpose of this paper is to extend such an approach to a wider class of dynamic systeIns,namely, genera1ized Hamiltonian systems. Our purpose is to investigate a class of dynaInicsystems, which possess a certain "geometric structure".Deflnition 1.1[1'2]. Let M be a tIlallifo1d. w E fl'(M) is call…     

Quantaloids Describing Causation and Propagation of Physical Properties

A general principle of causal duality for physical systems, lying at the base of representation theorems for both compound and evolving systems, is proved; formally it is encoded in a quantaloidal setting. Other particular examples of quantaloids and quantaloidal morphisms appear naturally within this setting; as in the case of causal duality, they originate from primitive physical reasonings on the lattices of properties of physical systems. Furthermore, an essentially dynamical operational foundation for studying physical systems is outlined; complementary as it is to the existing static operational foundation, it leads to the natural axiomatization of causal duality in operational quantum logic.     

On fillable contact structures up to homotopy

Let be a closed, oriented -manifold. The set of homotopy classes of positive, fillable contact structures on is a subtle invariant of , known to always be a finite set. In this paper we study under the assumption that carries metrics with positive scalar curvature. Using Seiberg-Witten gauge theory, we prove that two positive, fillable contact structures on are homotopic if and only if they are homotopic on the complement of a point. This implies that the cardinality of is bounded above by the order of the torsion subgroup of . Using explicit examples we show that without the geometric assumption on such a bound can be arbitrarily far from holding.

     

A Note on the KAM Theorem for Symplectic Mappings

The mapping version of Pöschel's theory on differentiable foliation structures of invariant tori is presented and the relevant estimates explicitly in terms of the diophantine constant and the nondegeneracy parameters of frequency maps are given. As a direct application of the main result, a generalization of Moser's small twist theorem to high dimensions is given.     

A Quantization on Riemann Surfaces with Projective Structure

Let X be a Riemann surface equipped with a projective structure. Let be a square-root of the holomorphic cotangent bundle K _X . Consider the symplectic form on the complement of the zero section of obtained by pulling back the symplectic form on K _X using the map ². We show that this symplectic form admits a natural quantization. This quantization also gives a quantization of the complement of the zero section in K _X equipped with the natural symplectic form.     

两个玻色-爱因斯坦凝聚体间相互作用的数值研究

本文采用辛算法数值求解一维含时Gross-Pitaevskii(GP)方程.研究了存在陷俘势和撤掉陷俘势时两个凝聚体间的相互作用.发现当存在陷俘势时两个凝聚体间发生弹性碰撞;在零时刻撤掉陷俘势时两个凝聚体间发生干涉现象;当t=2时撤掉陷俘势两个凝聚体间发生了复杂现象.