VARIATIONS ON A THEME BY EULER

1.IntroductionAHallliltolliansystemofdifferentialequationsonRZnisgivedbyP~~H,(P,q),q=HP(P,q),(1)wherep=(pl,'.,P.),q=(ql,',q.)eR"arethegeneralizedcoordinatesandmolllentarespectivelyandH(P,q)istheellergyofthesystem.Thesystem(1)canberewrittenasthecompactf…     

A particle method for a self-gravitating fluid: A convergence result

We study an Hamiltonian system of N particles in ?³ interacting by a short-range repulsive and a long-range attractive potential. It is shown that the empirical measures associated to the positions and velocity of the system converge to the solutions of Euler equations for a self-gravitating fluid, in the limit as the particle number tends to infinity, for a suitable scaling of the interactions.     

Poisson系统的辛结构

当Poisson系统中的Poisson矩阵是非常数时,经典的辛方法如辛Runge-Kutta方法,生成函数法一般不能保持Poisson系统的Poisson结构,利用非线性变换可把非常数Poisson结构转化成辛结构,然后任意阶的辛方法可以长时间计算Poisson系统的辛结构．自由刚体问题中Euler方程被转换成辛结构并用辛中点格式进行数值求解,数值结果给出了这种非线性变换的有效性．     

Global Symplectic Polynomial Approximation of Area-Preserving Maps

Different methods are proposed and tested for transforming a nonlinear differential system, and more particularly a Hamiltonian one, into a map without having to integrate the whole orbit as in the well known Poincaré map technique . We construct a synthetic map, based on a global fitting, which satisfies the symplectic condition. Taking the Standard Map as a model problem we point our attention on methods suitable for comparing the model map and its synthetic counterpart. We test the agreement of the fitting on finer scales through the visual representation, the computation of the rotation number and the measure of the local distribution of the Lyapunov characteristic exponents. Comparing these results with those obtained by Froeschlé and Petit using a method based on Taylor interpolation, we show that the symplectic character is a crucial condition for the recovering of the finest details of a dynamical system. On the other hand the global character of our method makes the generalization to any system of differential equations difficult.     

A vortex method for fluid-particle systems

We consider a two-dimensional, dilute fluid-particle system with low Reynolds number for the flow around the particles and high Reynolds number for the bulk flow. We use a vortex method to calculate numerically the incompressible fluid phase. For the compressible particle phase we use a particle method and Voronoi diagrams to calculate the particle density. We use the Stokes-Oseen formula to represent approximately the force of the fluid on the particles. We give the results of a numerical experiment that show the effect of fluid particle interaction on the bulk flow.     

Dynamics of symplectic fluids and point vortices

We present the Hamiltonian formalism for the Euler equation of symplectic fluids, introduce symplectic vorticity, and study related invariants. In particular, this allows one to extend Ebin’s long-time existence result for geodesics on the symplectomorphism group to metrics not necessarily compatible with the symplectic structure. We also study the dynamics of symplectic point vortices, describe their symmetry groups and integrability.     

A sixth-order dual preserving algorithm for the Camassa-Holm equation

The paper presents a sixth-order numerical algorithm for studying the completely integrable Camassa-Holm (CH) equation. The proposed sixth-order accurate method preserves both the dispersion relation and the Hamiltonians of the CH equation. The CH equation in this study is written as an evolution equation, involving only the first-order spatial derivatives, coupled with the Helmholtz equation. We propose a two-step method that first solves the evolution equation by a sixth-order symplectic Runge-Kutta method and then solves the Helmholtz equation using a three-point sixth-order compact scheme. The first-order derivative terms in the first step are approximated by a sixth-order dispersion-relation-preserving scheme that preserves the physically inherent dispersive nature. The compact Helmholtz solver, on the other hand, allows us to use relatively few nodal points in a stencil, while achieving a higher-order accuracy. The sixth-order symplectic Runge-Kutta time integrator is preferable for an equation that possesses a Hamiltonian structure. We illustrate the ability of the proposed scheme by examining examples involving peakon or peakon-like solutions. We compare the computed solutions with exact solutions or asymptotic predictions. We also demonstrate the ability of the symplectic time integrator to preserve the Hamiltonians. Finally, via a smooth travelling wave problem, we compare the accuracy, elapsed computing time, and rate of convergence among the proposed method, a second-order two-step algorithm, and a completely integrable particle method.     

Minimizing Euler characteristics of symplectic four-manifolds

We prove that the minimal Euler characteristic of a closed symplectic four-manifold with given fundamental group is often much larger than the minimal Euler characteristic of almost complex closed four-manifolds with the same fundamental group. In fact, the difference between the two is arbitrarily large for certain groups.

     

The distribution of the spine of a Fleming–Viot type process

We show uniqueness of the spine of a Fleming–Viot particle system under minimal assumptions on the driving process. If the driving process is a continuous time Markov process on a finite space, we show that asymptotically, when the number of particles goes to infinity, the distribution of the spine converges to that of the driving process conditioned to stay alive forever, the branching rate for the spine is twice that of a generic particle in the system, and every side branch has the distribution of the unconditioned generic branching tree.     

Circular Distributions and Euler Systems, II

The purpose of this paper is to investigate a conjecture about the universality of the circular distribution made by Robert Coleman. The algebraic property of the universal distribution is the main ingredient in studying Euler system of Kolyvagin and Rubin. We study the universality of the circular distribution by using the Iwasawa theory and the theory of the Euler systems. The conjecture is a characterization of Euler systems in the case of number field. The results here assert that Euler systems are essentially made out of cyclotomic units.     

Mean-field limit for the stochastic Vicsek model

We consider the continuous version of the Vicsek model with noise, proposed as a model for collective behaviour of individuals with a fixed speed. We rigorously derive the kinetic mean-field partial differential equation satisfied when the number N of particles tends to infinity, quantifying the convergence of the law of one particle to the solution of the PDE. For this we adapt a classical coupling argument to the present case in which both the particle system and the PDE are defined on a surface rather than on the whole space R^d. As part of the study we give existence and uniqueness results for both the particle system and the PDE.     

On a practical implementation of particle methods

This paper is devoted to a practical implementation of deterministic particle methods for solving transport equations with discontinuous coefficients and/or initial data, and related problems. In such methods, the solution is sought in the form of a linear combination of the delta-functions, whose positions and coefficients represent locations and weights of the particles, respectively. The locations and weights of the particles are then evolved in time according to a system of ODEs, obtained from the weak formulation of the transport PDEs.The major theoretical difficulty in solving the resulting system of ODEs is the lack of smoothness of its right-hand side. While the existence of a generalized solution is guaranteed by the theory of Filippov, the uniqueness can only be obtained via a proper regularization. Another difficulty one may encounter is related to an interpretation of the computed solution, whose point values are to be recovered from its particle distribution. We demonstrate that some of known recovering procedures, suitable for smooth functions, may fail to produce reasonable results in the nonsmooth case, and discuss several successful strategies which may be useful in practice. Different approaches are illustrated in a number of numerical examples, including one- and two-dimensional transport equations and the reactive Euler equations of gas dynamics.     

Geometric Integrators for Higher-Order Variational Systems and Their Application to Optimal Control

Numerical methods that preserve geometric invariants of the system, such as energy, momentum or the symplectic form, are called geometric integrators. In this paper we present a method to construct symplectic-momentum integrators for higher-order Lagrangian systems. Given a regular higher-order Lagrangian \(L:T^{(k)}Q\rightarrow {\mathbb {R}}\) with \(k\ge 1\), the resulting discrete equations define a generally implicit numerical integrator algorithm on \(T^{(k-1)}Q\times T^{(k-1)}Q\) that approximates the flow of the higher-order Euler–Lagrange equations for L. The algorithm equations are called higher-order discrete Euler–Lagrange equations and constitute a variational integrator for higher-order mechanical systems. The general idea for those variational integrators is to directly discretize Hamilton’s principle rather than the equations of motion in a way that preserves the invariants of the original system, notably the symplectic form and, via a discrete version of Noether’s theorem, the momentum map. We construct an exact discrete Lagrangian \(L_d^e\) using the locally unique solution of the higher-order Euler–Lagrange equations for L with boundary conditions. By taking the discrete Lagrangian as an approximation of \(L_d^e\), we obtain variational integrators for higher-order mechanical systems. We apply our techniques to optimal control problems since, given a cost function, the optimal control problem is understood as a second-order variational problem.     

Tamagawa defect of Euler systems

As remarked by Mazur and Rubin [B. Mazur, K. Rubin, Kolyvagin systems, Mem. Amer. Math. Soc. 168 (799) (2004)] one does not expect the Kolyvagin system obtained from an Euler system for a p-adic Galois representation T to be primitive (in the sense of the above mentioned reference) if p divides a Tamagawa number at a prime ?≠p; thus fails to compute the correct size of the relevant Selmer module. In this paper we obtain a lower bound for the size of the cokernel of the Euler system to Kolyvagin system map in terms of the local Tamagawa numbers of T, refining a result of [B. Mazur, K. Rubin, Kolyvagin systems, Mem. Amer. Math. Soc. 168 (799) (2004)]. We show how this partially accounts for the missing Tamagawa factors in Kato's calculations with his Euler system.     

Formulation,Stability, and Computation of Traffic Network Equilibria as Projected Dynamical Systems

In this paper, we present a unified treatment and analysis of a dynamic traffic network model with elastic demands formulated and studied as a projected dynamical system. We propose a travel route choice adjustment process that satisfies the projected dynamical system. Under certain conditions, stability and asymptotical stability of the equilibrium patterns are then derived. Finally, two discrete-time algorithms, the Euler method and the Heun method, are proposed for the computation of the solutions, and convergence results established. The convergence results depend crucially on stability analysis. The performance of the algorithms is then illustrated on several transportation networks.     

On a Convex Embedding of the Euler Problem of Two Fixed Centers

In this article, we study a convex embedding for the Euler problem of two fixed centers for energies below the critical energy level. We prove that the doubly-covered elliptic coordinates provide a 2-to-1 symplectic embedding such that the image of the bounded component near the lighter primary of the regularized Euler problem is convex for any energy below the critical Jacobi energy. This holds true if the two primaries have equal mass, but does not hold near the heavier body.     

转子动力学的非线性数值求解

将Euler(欧拉)角表示引入转子动力学系统,用以描述转子的非线性旋转运动,并与时间有限元相结合,进而提出了包含非线性因素的转子动力学保辛数值求解方法.以此方法为基础,分析了悬臂梁-圆盘转子系统的涡动行为.数值结果证明该数值解法的有效性与正确性,可用于各种转子系统涡动行为分析.     

Comparison of non-cohesive resolved and coarse grain DEM models for gas flow through particle beds

The Discrete Element Method (DEM) is a widely used approach for modelling granular systems. Currently, the number of particles which can be tractably modelled using DEM is several orders of magnitude lower than the number of particles present in common large-scale industrial systems. Practical approaches to modelling such industrial system therefore usually involve modelling over a limited domain, or with larger particle diameters and a corresponding assumption of scale invariance. These assumption are, however, problematic in systems where granular material interacts with gas flow, as the dynamics of the system depends heavily on the number of particles. This has led to a number of suggested modifications for coupled gas–grain DEM to effectively increase the number of particles being simulated. One such approach is for each simulated particle to represent a cluster of smaller particles and to re-formulate DEM based on these clusters. This, known as a representative or ‘coarse grain’ method, potentially allows the number of virtual DEM particles to be approximately the same as the real number of particles at relatively low computational cost. We summarise the current approaches to coarse grain models in the literature, with emphasis on discussion of limitations and assumptions inherent in such approaches. The effectiveness of the method is investigated for gas flow through particle beds using resolved and coarse grain models with the same effective particle numbers. The pressure drop, as well as the pre and post fluidisation characteristics in the beds are measured and compared, and the relative saving in computational cost is weighed against the effectiveness of the coarse grain approach. In general, the method is found perform reasonably well, with a considerable saving of computational time, but to deviate from empirical predictions at large coarse grain ratios.     

Theta functions on <Emphasis Type="Italic">T</Emphasis><Superscript>2</Superscript>-bundles over <Emphasis Type="Italic">T</Emphasis><Superscript>2</Superscript> with the zero Euler class

We construct an analog of the classical theta function on an abelian variety for the closed 4-dimensional symplectic manifolds that are T ²-bundles over T ² with the zero Euler class. We use our theta functions for a canonical symplectic embedding of these manifolds into complex projective spaces (an analog of the Lefschetz theorem).     

Two-phase flow simulation for distinguishing deformable particles with a LiMCA system

In the metallurgical industry, Liquid Metal Cleanliness Analyser (LiMCA) commercial equipment cannot distinguish between hard particles (e.g., oxides, borides) and deformable particles (e.g., bubbles, molten salts). Therefore, hard particle concentrations can sometimes be grossly overestimated, which reduces the measurement accuracy. However, the method could potentially discriminate between deformable particles and hard particles by evaluating the particle's ability to deform. In this work, the coupled multiphysics problem of a particle deforming within current-carrying aluminium metal passing through the electric sensing zone (ESZ) is simulated using the conservative level-set (CLS) method. An emphasis is placed on understanding the transient deformation history, and the effect of the capillary number, Reynolds number, and confinement ratio on deformation are studied. Furthermore, a computational basis is given to estimate the influence of particle deformation on electrical resistance pulses (ERP). It is found that ERP features of deformation particles, including the peak magnitude and the pulse width, are different from those of hard particles. Based on the results, the effect of a particle's deformation and the feasibility to discriminate it from non-deformable particles in the LiMCA system is evaluated.