Symmetric and symplectic exponential integrators for nonlinear Hamiltonian systems

This letter studies symmetric and symplectic exponential integrators when applied to numerically computing nonlinear Hamiltonian systems. We first establish the symmetry and symplecticity conditions of exponential integrators and then show that these conditions are extensions of the symmetry and symplecticity conditions of Runge–Kutta methods. Based on these conditions, some symmetric and symplectic exponential integrators up to order four are derived. Two numerical experiments are carried out and the results demonstrate the remarkable numerical behaviour of the new exponential integrators in comparison with some symmetric and symplectic Runge–Kutta methods in the literature.     

High-Order Symplectic and Symmetric Composition Methods for Multi-Frequency and Multi-Dimensional Oscillatory Hamiltonian Systems

The multi-frequency and multi-dimensional adapted Runge-Kutta-Nyström (ARKN) integrators, and multi-frequency and multi-dimensional extended Runge-Kutta-Nyström(ERKN) integrators have been developed to efficiently solve multi-frequency oscillatory Hamiltonian systems. The aim of this paper is to analyze and derive high-order symplectic and symmetric composition methods based on the ARKN integrators and ERKN integrators. We first consider the symplecticity conditions for the multi-frequency and multi-dimensional ARKN integrators. We then analyze the symplecticity of the adjoint integrators of the multi-frequency and multi-dimensional symplectic ARKN integrators and ERKN integrators, respectively. On the basis of the theoretical analysis and by using the idea of composition methods, we derive and propose four new high-order symplectic and symmetric methods for the multi-frequency oscillatory Hamiltonian systems. The numerical results accompanied in this paper quantitatively show the advantage and efficiency of the proposed high-order symplectic and symmetric methods.     

Hamilton–Pontryagin Integrators on Lie Groups Part I: Introduction and Structure-Preserving Properties

In this paper, structure-preserving time-integrators for rigid body-type mechanical systems are derived from a discrete Hamilton–Pontryagin variational principle. From this principle, one can derive a novel class of variational partitioned Runge–Kutta methods on Lie groups. Included among these integrators are generalizations of symplectic Euler and Störmer–Verlet integrators from flat spaces to Lie groups. Because of their variational design, these integrators preserve a discrete momentum map (in the presence of symmetry) and a symplectic form. In a companion paper, we perform a numerical analysis of these methods and report on numerical experiments on the rigid body and chaotic dynamics of an underwater vehicle. The numerics reveal that these variational integrators possess structure-preserving properties that methods designed to preserve momentum (using the coadjoint action of the Lie group) and energy (for example, by projection) lack.     

A long-term numerical energy-preserving analysis of symmetric and/or symplectic extended RKN integrators for efficiently solving highly oscillatory Hamiltonian systems

This paper presents a long-term analysis of one-stage extended Runge–Kutta–Nyström (ERKN) integrators for highly oscillatory Hamiltonian systems. We study the long-time numerical energy conservation not only for symmetric integrators but also for symplectic integrators. In the analysis, we neither assume symplecticity for symmetric methods, nor assume symmetry for symplectic methods. It turns out that these both types of integrators have a near conservation of the total and oscillatory energy over a long term. To prove the result for explicit integrators, a relationship between ERKN integrators and trigonometric integrators is established. For the long-term analysis of implicit integrators, the above approach does not work anymore and we use the technology of modulated Fourier expansion. By taking some adaptations of this technology for implicit methods, we derive the modulated Fourier expansion and show the near energy conservation.

     

Regular reduction of controlled Hamiltonian system with symplectic structure and symmetry

In this paper, our goal is to study the regular reduction theory of regular controlled Hamiltonian (RCH) systems with symplectic structure and symmetry, and this reduction is an extension of regular symplectic reduction theory of Hamiltonian systems under regular controlled Hamiltonian equivalence conditions. Thus, in order to describe uniformly RCH systems defined on a cotangent bundle and on the regular reduced spaces, we first define a kind of RCH systems on a symplectic fiber bundle. Then we introduce regular point and regular orbit reducible RCH systems with symmetry by using momentum map and the associated reduced symplectic forms. Moreover, we give regular point and regular orbit reduction theorems for RCH systems to explain the relationships between RpCH-equivalence, RoCH-equivalence for reducible RCH systems with symmetry and RCH-equivalence for associated reduced RCH systems. Finally, as an application we regard rigid body and heavy top as well as them with internal rotors as the regular point reducible RCH systems on the rotation group SO(3) and on the Euclidean group SE(3), as well as on their generalizations, respectively, and discuss their RCH-equivalence. We also describe the RCH system and RCH-equivalence from the viewpoint of port Hamiltonian system with a symplectic structure.     

New open modified Newton Cotes type formulae as multilayer symplectic integrators

New modified open Newton Cotes integrators are introduced in this paper. For the new proposed integrators the connection between these new algorithms, differential methods and symplectic integrators is studied. Much research has been done on one step symplectic integrators and several of them have obtained based on symplectic geometry. However, the research on multistep symplectic integrators is very poor. Zhu et al. [1] studied the well known open Newton Cotes differential methods and they presented them as multilayer symplectic integrators. Chiou and Wu [2] studied the development of multistep symplectic integrators based on the open Newton Cotes integration methods. In this paper we introduce a new open modified numerical method of Newton Cotes type and we present it as symplectic multilayer structure. The new obtained symplectic schemes are applied for the solution of Hamilton’s equations of motion which are linear in position and momentum. An important remark is that the Hamiltonian energy of the system remains almost constant as integration proceeds. We have applied also efficiently the new proposed method to a nonlinear orbital problem and an almost periodic orbital problem.     

Geometric integration of non-autonomous linear Hamiltonian problems

Symplectic integration of autonomous Hamiltonian systems is a well-known field of study in geometric numerical integration, but for non-autonomous systems the situation is less clear, since symplectic structure requires an even number of dimensions. We show that one possible extension of symplectic methods in the autonomous setting to the non-autonomous setting is obtained by using canonical transformations. Many existing methods fit into this framework. We also perform experiments which indicate that for exponential integrators, the canonical and symmetric properties are important for good long time behaviour. In particular, the theoretical and numerical results support the well documented fact from the literature that exponential integrators for non-autonomous linear problems have superior accuracy compared to general ODE schemes.     

关于辛算法稳定性的若干注记

我们讨论辛算法的线性稳定性和非线性稳定性,从动力系统和计算的角度论述了研究辛算法的这两类稳定性问题的重要性,分析总结了相关重要结果.我们给出了解析方法的明确定义,证明了稳定函数是亚纯函数的解析辛方法是绝对线性稳定的.绝对线性稳定的辛方法既有解析方法（如Runge-Kutta辛方法）,也有非解析方法（如基于常数变易公式对线性部分进行指数积分而对非线性部分使用其它数值积分的方法）.我们特别回顾并讨论了R.I.McLachlan,S.K.Gray和S.Blanes,F.Casas,A.Murua等关于分裂算法的线性稳定性结果,如通过选取适当的稳定多项式函数构造具有最优线性稳定性的任意高阶分裂辛算法和高效共轭校正辛算法,这类经优化后的方法应用于诸如高振荡系统和波动方程等线性方程或者线性主导的弱非线性方程具有良好的数值稳定性.我们通过分析辛算法在保持椭圆平衡点的稳定性,能量面的指数长时间慢扩散和KAM不变环面的保持等三个方面阐述了辛算法的非线性稳定性,总结了相关已有结果.最后在向后误差分析基础上,基于一个自由度的非线性振子和同宿轨分析法讨论了辛算法的非线性稳定性,提出了一个新的非线性稳定性概念,目的是为辛算法提供一个实际可用的非线性稳定性判别法.     

Minimal surfaces and particles in 3-manifolds

We consider 3-dimensional anti-de Sitter manifolds with conical singularities along time-like lines, which is what in the physics literature is known as manifolds with particles. We show that the space of such cone-manifolds is parametrized by the cotangent bundle of Teichmüller space, and that moreover such cone-manifolds have a canonical foliation by space-like surfaces. We extend these results to de Sitter and Minkowski cone-manifolds, as well as to some related “quasifuchsian” hyperbolic manifolds with conical singularities along infinite lines, in this later case under the condition that they contain a minimal surface with principal curvatures less than 1. In the hyperbolic case the space of such cone-manifolds turns out to be parametrized by an open subset in the cotangent bundle of Teichmüller space. For all settings, the symplectic form on the moduli space of 3-manifolds that comes from parameterization by the cotangent bundle of Teichmüller space is the same as the 3-dimensional gravity one. The proofs use minimal (or maximal, or CMC) surfaces, along with some results of Mess on AdS manifolds, which are recovered here in a different way, using differential-geometric methods and a result of Labourie on some mappings between hyperbolic surfaces, that allows an extension to cone-manifolds.      

Closed Newton–Cotes trigonometrically-fitted formulae of high order for long-time integration of orbital problems

The connection between closed Newton–Cotes, trigonometrically-fitted differential methods and symplectic integrators is studied in this paper. Several one-step symplectic integrators have been obtained based on symplectic geometry, as is shown in the literature. However, the study of multi-step symplectic integrators is very limited. The well-known open Newton–Cotes differential methods are presented as multilayer symplectic integrators by Zhu et al. [W. Zhu, X. Zhao, Y. Tang, Journal of Chem. Phys. 104 (1996), 2275]. The construction of multi-step symplectic integrators based on the open Newton–Cotes integration methods is investigated by Chiou and Wu [J.C. Chiou, S.D. Wu, Journal of Chemical Physics 107 (1997), 6894]. The closed Newton–Cotes formulae are studied in this paper and presented as symplectic multilayer structures. We also develop trigonometrically-fitted symplectic methods which are based on the closed Newton–Cotes formulae. We apply the symplectic schemes in order to solve Hamilton’s equations of motion which are linear in position and momentum. We observe that the Hamiltonian energy of the system remains almost constant as the integration proceeds. Finally we apply the new developed methods to an orbital problem in order to show the efficiency of this new methodology.     

The symplectic normal space of a cotangent-lifted action

For the cotangent bundle T^∗Q of a smooth Riemannian manifold acted upon by the lift of a smooth and proper action by isometries of a Lie group, we characterize the symplectic normal space at any point. We show that this space splits as the direct sum of the cotangent bundle of a linear space and a symplectic linear space coming from reduction of a coadjoint orbit. This characterization of the symplectic normal space can be expressed solely in terms of the group action on the base manifold and the coadjoint representation. Some relevant particular cases are explored.     

Higher order variational integrators in optimal control theory

Higher order variational integrators are analyzed and applied to optimal control problems posed with mechanical systems. First, we derive two different kinds of high order variational integrators based on different dimensions of the underlying approximation space. While the first well-known integrator is equivalent to a symplectic partitioned Runge-Kutta method, the second integrator, denoted as symplectic Galerkin integrator, yields a method which in general, cannot be written as a standard symplectic Runge-Kutta scheme [1]. Furthermore, we use these integrators for the discretization of optimal control problems. By analyzing the adjoint systems of the optimal control problem and its discretized counterpart, we prove that for these particular integrators optimization and discretization commute [2]. This property guarantees that the accuracy is preserved for the adjoint system which is also referred to as the Covector Mapping Principle. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Singular cotangent bundle reduction & spin Calogero–Moser systems

We develop a bundle picture for singular symplectic quotients of cotangent bundles acted upon by cotangent lifted actions for the case that the configuration manifold is of single orbit type. Furthermore, we give a formula for the reduced symplectic form in this setting. As an application of this bundle picture we consider Calogero–Moser systems with spin associated to polar representations of compact Lie groups.     

Stochastic discrete Hamiltonian variational integrators

Variational integrators are derived for structure-preserving simulation of stochastic Hamiltonian systems with a certain type of multiplicative noise arising in geometric mechanics. The derivation is based on a stochastic discrete Hamiltonian which approximates a type-II stochastic generating function for the stochastic flow of the Hamiltonian system. The generating function is obtained by introducing an appropriate stochastic action functional and its corresponding variational principle. Our approach permits to recast in a unified framework a number of integrators previously studied in the literature, and presents a general methodology to derive new structure-preserving numerical schemes. The resulting integrators are symplectic; they preserve integrals of motion related to Lie group symmetries; and they include stochastic symplectic Runge–Kutta methods as a special case. Several new low-stage stochastic symplectic methods of mean-square order 1.0 derived using this approach are presented and tested numerically to demonstrate their superior long-time numerical stability and energy behavior compared to nonsymplectic methods.     

Volume preserving RK methods for linear systems

1. IntroductionIn recent yearss there has been a great interest in constructing numerical integrationschemes for ODEs in such a way that some qualitative geometrical properties of the solutionof the ODEs are exactly preserved. R.th[ll and Feng Kang[2'31 has proposed symplectic algorithms for Hamiltollian systems, and since then st ruct ure s- preserving me t ho ds fordynamical systems have been systematically developed[4--7]. The symplectic algorithms forHamiltonian systems, the volume-pre…     

基于辛格式的深度哈密尔顿神经网络

HNN是一类基于物理先验学习哈密尔顿系统的神经网络.本文通过误差分析解释使用不同积分器作为超参数对HNN的影响.如果我们把网络目标定义为在任意训练集上损失为零的映射,那么传统的积分器无法保证HNN存在网络目标.我们引进反修正方程,并严格证明基于辛格式的HNN具有网络目标,且它与原哈密尔顿量之差依赖于数值格式的精度.数值实验表明,由辛HNN得到的哈密尔顿系统的相流不能精确保持原哈密尔顿量,但保持网络目标;网络目标在训练集、测试集上的损失远小于原哈密尔顿量的损失;在预测问题上辛HNN较非辛HNN具备更强大的泛化能力和更高的精度.因此,辛格式对于HNN是至关重要的.     

Essentially Symplectic Boundary Value Methods for Linear Hamiltonian Systems

1.IlltroductiollInmanyareasofphysics,mechanics,etc.,HamiltoniansystemsofODEsplayaveryimportantrole.Suchsystemshavethefollowinggeneralform:where,bydenotingwithOfandimthenullmatrixandtheidentitymatrixofordermarespectively,SimplepropertiesofthematrixJZmarethefollowingones:Inequation(1)AH(~,t)isthegradientofascalarfunctionH(y,t),usuallycalledHamiltonian.InthecasewhereH(y,t)=H(y),thenthevalueofthisfunctionremainsconstantalongt.hesollltion7/(t),t,hatis'*ReceivedFebruaryI3,1995.l)Worksupporte…     

Hamiltonian reduction of free particle motion on the group SL(2, ℝ)

We analyze the structure of the reduced phase space that arises in the Hamiltonian reduction of the phase space of free particle motion over the groupSL(2, ℝ). The reduction considered is based on introducing constraints that are analogous to those used in the reduction of the Wess-Zumino-Novikov-Witten model to Toda systems. It is shown that the reduced phase space is diffeomorphic either to a union of two two-dimensional planes or to a cylinder S¹×ℝ. We construct canonical coordinates for both cases and show that in the first case, the reduced phase space is symplectomorphic to the union of two cotangent bundles T*(ℝ) endowed with a canonical symplectic structure, while in the second case, it is symplectomorphic to the cotangent bundle T* (S¹), which is also endowed with a canonical symplectic structure. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 110, No. 1, pp. 149–161, January, 1997.