大型不正定陀螺系统本征值问题

陀螺动力系统可以导入哈密顿辛几何体系,在哈密顿陀螺系统的辛子空间迭代法的基础上提出了一种能够有效计算大型不正定哈密顿函数的陀螺系统本征值问题的算法．利用陀螺矩阵既为哈密顿矩阵而本征值又是纯虚数或零的特点,将对应哈密顿函数为负的本征值分离开来,构造出对应哈密顿函数全为正的本征值问题,利用陀螺系统的辛子空间迭代法计算出正定哈密顿矩阵的本征值,从而解决了大型不正定陀螺系统的本征值问题,算例证明,本征解收敛得很快．     

Hamiltonian spatial structure for three-dimensional water waves in a moving frame of reference

Summary The governing equations for three-dimensional time-dependent water waves in a moving frame of reference are reformulated in terms of the energy and momentum flux. The novelty of this approach is that time-independent motions of the system—that is, motions that are steady in a moving frame of reference—satisfy a partial differential equation, which is shown to be Hamiltonian. The theory of Hamiltonian evolution equations (canonical variables, Poisson brackets, symplectic form, conservation laws) is applied to the spatial Hamiltonian system derived for pure gravity waves. The addition of surface tension changes the spatial Hamiltonian structure in such a way that the symplectic operator becomes degenerate, and the properties of this generalized Hamiltonian system are also studied. Hamiltonian bifurcation theory is applied to the linear spatial Hamiltonian system for capillary-gravity waves, showing how new waves can be found in this framework.     

On averaging,reduction, and symmetry in hamiltonian systems

The existence of periodic orbits for Hamiltonian systems at low positive energies can be deduced from the existence of nondegenerate critical points of an averaged Hamiltonian on an associated “reduced space.” Alternatively, in classical (kinetic plus potential energy) Hamiltonians the existence of such orbits can often be established by elementary geometrical arguments. The present paper unifies the two approaches by exploiting discrete symmetries, including reversing diffeomorphisms, that occur in a given system. The symmetries are used to locate the periodic orbits in the averaged Hamiltonian, and thence in the original Hamiltonian when the periodic orbits are continued under perturbations admitting the same symmetries. In applications to the Hénon-Heiles Hamiltonian, it is illustrated how “higher order” averaging can sometimes be used to overcome degeneracies encountered at first order.     

A boundary layer theory for diffusively perturbed transport around a heteroclinic cycle

We consider a two‐dimensional transport equation subject to small diffusive perturbations. The transport equation is given by a Hamiltonian flow near a compact and connected heteroclinic cycle. We investigate approximately harmonic functions corresponding to the generator of the perturbed transport equation. In particular, we investigate such functions in the boundary layer near the heteroclinic cycle; the space of these functions gives information about the likelihood of a particle moving a mesoscopic distance into one of the regions where the transport equation corresponds to periodic oscillations (i.e., a “well” of the Hamiltonian). We find that we can construct such approximately harmonic functions (which can be used as “corrector functions” in certain averaging questions) when certain macroscopic “gluing conditions” are satisfied. This provides a different perspective on some previous work of Freidlin and Wentzell on stochastic averaging of Hamiltonian systems. © 2004 Wiley Periodicals, Inc.     

The Hamiltonian connectivity of rectangular supergrid graphs

A Hamiltonian path of a graph is a simple path which visits each vertex of the graph exactly once. The Hamiltonian path problem is to determine whether a graph contains a Hamiltonian path. A graph is called Hamiltonian connected if there exists a Hamiltonian path between any two distinct vertices. In this paper, we will study the Hamiltonian connectivity of rectangular supergrid graphs. Supergrid graphs were first introduced by us and include grid graphs and triangular grid graphs as subgraphs. The Hamiltonian path problem for grid graphs and triangular grid graphs was known to be NP-complete. Recently, we have proved that the Hamiltonian path problem for supergrid graphs is also NP-complete. The Hamiltonian paths on supergrid graphs can be applied to compute the stitching traces of computer sewing machines. Rectangular supergrid graphs form a popular subclass of supergrid graphs, and they have strong structure. In this paper, we provide a constructive proof to show that rectangular supergrid graphs are Hamiltonian connected except one trivial forbidden condition. Based on the constructive proof, we present a linear-time algorithm to construct a longest path between any two given vertices in a rectangular supergrid graph.     

The existence of Hamiltonian stationary Lagrangian tori in K?hler manifolds of any dimension

Hamiltonian stationary Lagrangians are Lagrangian submanifolds that are critical points of the volume functional under Hamiltonian deformations. They are natural generalizations of special Lagrangians or Lagrangian and minimal submanifolds. In this paper, we obtain a local condition that gives the existence of a smooth family of Hamiltonian stationary Lagrangian tori in K?hler manifolds. This criterion involves a weighted sum of holomorphic sectional curvatures. It can be considered as a complex analogue of the scalar curvature when the weighting are the same. The problem is also studied by Butscher and Corvino (Hamiltonian stationary tori in Kahler manifolds, 2008).     

A multisymplectic framework for classical field theory and the calculus of variations II: space + time decomposition

In a previous paper I laid the foundations of a covariant Hamiltonian framework for the calculus of variations in general. The purpose of the present work is to demonstrate, in the context of classical field theory, how this covariant Hamiltonian formalism may be space + time decomposed. It turns out that the resulting “instantaneous” Hamiltonian formalism is an infinite- dimensional version of Ostrogradski 's theory and leads to the standard symplectic formulation of the initial value problem. The salient features of the analysis are: (i) the instantaneous Hamiltonian formalism does not depend upon the choice of Lepagean equivalent; (ii) the space + time decomposition can be performed either before or after the covariant Legendre transformation has been carried out, with equivalent results; (iii) the instantaneous Hamiltonian can be recovered in natural way from the multisymplectic structure inherent in the theory; and (iv) the space + time split symplectic structure lives on the space of Cauchy data for the evolution equations, as opposed to the space of solutions thereof.     

Symplectic Transformations and a Reduction Method for Asymptotically Linear Hamiltonian Systems

In many fields of applications, especially in applications from mechanics, many equations of motion can be written as Hamiltonian systems. In this paper, we study a class of asymptotically linear Hamiltonian systems. We construct a symplectic transformation which reduces the linear systems of the Hamiltonian systems. This reduction method can be applied to study the existence of periodic solutions for a class of asymptotically linear Hamiltonian systems under weaker conditions on the linear systems of the Hamiltonian systems.     

A Generalized W-Transformation for Constructing Symplectic Partitioned Runge-Kutta Methods

This paper deals with the construction of implicit symplectic partitioned Runge–Kutta methods (PRKM) of high order for separable and general partitioned Hamiltonian systems. The main tool is a generalized W-transformation for PRKM based on different quadrature formulas. Methods of high order and special properties can be determined using the transformed coefficient matrices. Examples are given.     

A KAM-type Theorem for Generalized Hamiltonian Systems

In this paper we mainly concern the persistence of invariant tori in generalized Hamiltonian systems. Here the generalized Hamiltonian systems refer to the systems which may admit a distinct number of action and angle variables. In particular, system under consideration can be odd dimensional. Under the Riissmann type non-degenerate condition, we proved that the majority of the lower-dimension invariant tori of the integrable systems in generalized Hamiltonian system are persistent under small perturbation. The surviving lower-dimensional tori might be elliptic, hyperbolic, or of mixed type.     

Analytical bending solutions of clamped rectangular thin plates resting on elastic foundations by the symplectic superposition method

In this paper, the analytical bending solutions of clamped rectangular thin plates resting on elastic foundations are obtained by a rational symplectic superposition method which is based on the Hamiltonian system. The proposed method is capable of solving the plate problems with different boundary conditions via a step-by-step derivation without any trial solutions. The presented solution procedure can be extended to more boundary value problems in engineering.     

VON NEUMANN STABILITY ANALYSIS OF SYMPLECTIC INTEGRATORS APPLIED TO HAMILTONIAN PDEs

Symplectic integration of separable Hamiltonian ordinary and partial differential equations is discussed. A von Neumann analysis is performed to achieve general linear stability criteria for symplectic methods applied to a restricted class of Hamiltonian PDEs. In this treatment, the symplectic step is performed prior to the spatial step, as opposed to the standard approach of spatially discretising the PDE to form a system of Hamiltonian ODEs to which a symplectic integrator can be applied. In this way stability criteria are achieved by considering the spectra of linearised Hamiltonian PDEs rather than spatial step size.     

Lie algebras and Lie super algebra for the integrable couplings of NLS–MKdV hierarchy

Lie algebras and Lie super algebra are constructed and integrable couplings of NLS–MKdV hierarchy are obtained. Furthermore, its Hamiltonian and Super-Hamiltonian are presented by using of quadric-form identity and super-trace identity. The method can be used to produce the Hamiltonian structures of the other integrable and super-integrable systems.     

Method to derive Lagrangian and Hamiltonian for a nonlinear dynamical system with variable coefficients

A general method is developed to derive a Lagrangian and Hamiltonian for a nonlinear system with a quadratic first-order time derivative term and coefficients varying in the space coordinates. The method is based on variable transformations that allow removing the quadratic term and writing the equation of motion in standard form. Based on this form, an auxiliary Lagrangian for the transformed variables is derived and used to obtain the Lagrangian and Hamiltonian for the original variables. An interesting result is that the obtained Lagrangian and Hamiltonian can be non-local quantities, which do not diverge as the system evolves in time. Applications of the method to several systems with different coefficients shows that the method may become an important tool in studying nonlinear dynamical systems with a quadratic velocity term.     

Symplectic Difference Schemes for Linear Hamiltonian Canonical Systems

In this paper, we present some results of a study, specifically within the framework of symplectic geometry, of difference schemes for numerical solution of the linear Hamiltonian systems. We generalize the Cayley transform with which we can get different types of symplectic schemes. These schemes are various generalizations of the Euler centered scheme. They preserve all the invariant first integrals of the linear Hamiltonian systems.     

Construction of hamiltonian graphs and bigraphs with prescribed degrees

In this paper, we establish a laying-off algorithm for constructing a (simple) graph with prescribed degrees and with a Hamiltonian path starting at a specified vertex. This algorithm can be applied to construct a Hamiltonian graph with prescribed degrees. It is also shown that Hamiltonian bigraphs with prescribed degrees can be similarly constructed.     

Computing Cartograms with Optimal Complexity

In a rectilinear dual of a planar graph vertices are represented by simple rectilinear polygons, while edges are represented by side-contact between the corresponding polygons. A rectilinear dual is called a cartogram if the area of each region is equal to a pre-specified weight. The complexity of a cartogram is determined by the maximum number of corners (or sides) required for any polygon. In a series of papers the polygonal complexity of such representations for maximal planar graphs has been reduced from the initial 40 to 34, then to 12 and very recently to the currently best known 10. Here we describe a construction with 8-sided polygons, which is optimal in terms of polygonal complexity as 8-sided polygons are sometimes necessary. Specifically, we show how to compute the combinatorial structure and how to refine it into an area-universal rectangular layout in linear time. The exact cartogram can be computed from the area-universal layout with numerical iteration, or can be approximated with a hill-climbing heuristic. We also describe an alternative construction of cartograms for Hamiltonian maximal planar graphs, which allows us to directly compute the cartograms in linear time. Moreover, we prove that even for Hamiltonian graphs 8-sided rectilinear polygons are necessary, by constructing a non-trivial lower bound example. The complexity of the cartograms can be reduced to 6 if the Hamiltonian path has the extra property that it is one-legged, as in outer-planar graphs. Thus, we have optimal representations (in terms of both polygonal complexity and running time) for Hamiltonian maximal planar and maximal outer-planar graphs. Finally we address the problem of constructing small-complexity cartograms for 4-connected graphs (which are Hamiltonian). We first disprove a conjecture, posed by two set of authors, that any 4-connected maximal planar graph has a one-legged Hamiltonian cycle, thereby invalidating an attempt to achieve a polygonal complexity 6 in cartograms for this graph class. We also prove that it is NP-hard to decide whether a given 4-connected plane graph admits a cartogram with respect to a given weight function.     

VARIATIONAL INTEGRATORS FOR HIGHER ORDER DIFFERENTIAL EQUATIONS

We analyze three one parameter families of approximations and show that they are sympectic in Largrangian sence and can be related to symplectic schemes in Hamiltonian sense by different symplectic mapping.We also give a direct generalization of Veselov variational principlc for construction of scheme of higher order differential equations.At last,we present numerical experiments.