Reduced spaces and their relation to relative equilibria

Summary It is known that the Hamiltonian motion of a mechanical system with symmetry induces Hamiltonian flows on reduced phase spaces. In this paper we apply Morse theory to study the relationship between the topology of the reduced space and the number of relative equilibria in the corresponding momentum level set. Our attention is restricted to simple mechanical systems with compact configuration space and compact symmetry group. We begin by showing that the set of relative equilibria in a level set of the momentum map is compact. We then employ techniques from Morse theory to prove that the number of orbits of relative equilibria with momentum in the coadjoint orbit of a given regular momentum value is bounded below by the the sum of Betti numbers of the corresponding reduced space when the Hamiltonian is fibre quadratic and the reduced Hamiltonian is nondegenerate. In addition, for a certain class of group actions on the configuration manifold, it is shown that the above result extends to Hamiltonians of the form potential plus kinetic.     

双荷子系统运动的经典解

根据力学理论和经典电磁理论研究双荷子系统的运动.列出双荷子系统的运动微分方程,导出运动积分,说明系统的对称性,包括SO(4)对称性;利用变分法逆问题方法,构造双荷子系统的Lagrange(拉格朗日)函数和Hamilton(哈密顿)函数;解出双荷子系统的运动规律.     

Planar motion and stability for a rigid body with a beam in a field of central gravitational force

Planar motion for a rigid body with an elastic beam in a field of central gravitational force was investigated, and both of the orbital motion and attitude motion were under consideration. The equations of motion of the system were derived by the variational principle, and on view point of generalized Hamiltonian dynamics, the sufficient conditions for the stability of one class of relative equilibria were given by the energymomentum method.     

一类Hamilton系统的同宿解存在性

We study the existence of homoclinic orbits for some Hamiltonian system.A homoclinic orbit is obtained as a limit of 2kT-periodic solutions of a sequence of systems of differential equations.     

Normal forms and unfoldings of linear systems in eigenspaces of (anti)-automorphisms of order two

In this article we classify normal forms and unfoldings of linear maps in eigenspaces of (anti)-automorphisms of order two. Our main motivation is provided by applications to linear systems of ordinary differential equations, general and Hamiltonian, which have both time-preserving and time-reversing symmetries. However, the theory gives a uniform method to obtain normal forms and unfoldings for a wide variety of linear differential equations with additional structure. We give several examples and include a discussion of the phenomenon of orbit splitting. As a consequence of orbit splitting we observe passing and splitting of eigenvalues in unfoldings.     

Analytic and algebraic conditions for bifurcations of homoclinic orbits I: Saddle equilibria

We study bifurcations of homoclinic orbits to hyperbolic saddle equilibria in a class of four-dimensional systems which may be Hamiltonian or not. Only one parameter is enough to treat these types of bifurcations in Hamiltonian systems but two parameters are needed in general systems. We apply a version of Melnikov?s method due to Gruendler to obtain saddle-node and pitchfork types of bifurcation results for homoclinic orbits. Furthermore we prove that if these bifurcations occur, then the variational equations around the homoclinic orbits are integrable in the meaning of differential Galois theory under the assumption that the homoclinic orbits lie on analytic invariant manifolds. We illustrate our theories with an example which arises as stationary states of coupled real Ginzburg–Landau partial differential equations, and demonstrate the theoretical results by numerical ones.     

Runge-kutta schemes for Hamiltonian systems

We study the application of Runge-Kutta schemes to Hamiltonian systems of ordinary differential equations. We investigate which schemes possess the canonical property of the Hamiltonian flow. We also consider the issue of exact conservation in the time-discretization of the continuous invariants of motion.     

Qualitative analysis of systems with an ideal non-conservative constraint

The Hamiltonian form developed in /1/ for the equations of motion of systems with ideal non-conservative constraints enables familiar methods of classical and celestial mechanics to be used to analyse the dynamics of such systems. When this is done certain difficulties arise, due to the fact that the Hamiltonian is not analytic. In this paper one of the possible algorithms applying KAM theory /2/ and Poincaré's theory of periodic motions /3/ to the analysis of systems in which the Hamiltonian is non-analytic in one of the phase variables is described. As an example, some results of /4/ concerning the dynamics of a rigid body colliding with a fixed, absolutely smooth, horizontal plane are refined.     

Nonlinearization of the Lax pairs for discrete Ablowitz-Ladik hierarchy

The discrete Ablowitz-Ladik hierarchy with four potentials and the Hamiltonian structures are derived. Under a constraint between the potentials and eigenfunctions, the nonlinearization of the Lax pairs associated with the discrete Ablowitz-Ladik hierarchy leads to a new symplectic map and a class of finite-dimensional Hamiltonian systems. The generating function of the integrals of motion is presented, by which the symplectic map and these finite-dimensional Hamiltonian systems are further proved to be completely integrable in the Liouville sense. Each member in the discrete Ablowitz-Ladik hierarchy is decomposed into a Hamiltonian system of ordinary differential equations plus the discrete flow generated by the symplectic map.     

Orbital stability analysis using first integrals

A method is proposed for investigating the oribtal stability of periodic solutions of normal systems of ordinary differential equations. The Lyapunov function is derived from the first integrals of the equations of the perturbed motion and the scalar product of the velocity of motion along the orbit and the perturbation vector. Lypunov's second method was first used in connection with orbital stability in order to study the phase trajectories of systems with two degrees of freedom /1/.     

Relative equilibria of Hamiltonian systems with symmetry: Linearization,smoothness, and drift

Summary A relative equilibrium of a Hamiltonian system with symmetry is a point of phase space giving an evolution which is a one-parameter orbit of the action of the symmetry group of the system. The evolutions of sufficiently small perturbations of a formally stable relative equilibrium are arbitrarily confined to that relative equilibrium's orbit under the isotropy subgroup of its momentum. However, interesting evolution along that orbit, here called drift, does occur. In this article, linearizations of relative equilibria are used to construct a first order perturbation theory explaining drift, and also to determine when the set of relative equilibria near a given relative equilibrium is a smooth symplectic submanifold of phase space.     

Energy drift in molecular dynamics simulations

In molecular dynamics, Hamiltonian systems of differential equations are numerically integrated using the Störmer–Verlet method. One feature of these simulations is that there is an unphysical drift in the energy of the system over long integration periods. We study this energy drift, by considering a representative system in which it can be easily observed and studied. We show that if the system is started in a random initial configuration, the error in energy of the numerically computed solution is well modeled as a continuous-time stochastic process: geometric Brownian motion. We discuss what in our model is likely to remain the same or to change if our approach is applied to more realistic molecular dynamics simulations.     

Continuous-time limit of repeated interactions for a system in a confining potential

We study the continuous-time limit of a class of Markov chains coming from the evolution of classical open systems undergoing repeated interactions. This repeated interaction model has been initially developed for dissipative quantum systems in Attal and Pautrat (2006) and was recently set up for the first time in Deschamps (2012) for classical dynamics. It was particularly shown in the latter that this scheme furnishes a new kind of Markovian evolutions based on Hamilton’s equations of motion. The system is also proved to evolve in the continuous-time limit with a stochastic differential equation. We here extend the convergence of the evolution of the system to more general dynamics, that is, to more general Hamiltonians and probability measures in the definition of the model. We also present a natural way to directly renormalize the initial Hamiltonian in order to obtain the relevant process in a study of the continuous-time limit. Then, even if Hamilton’s equations have no explicit solution in general, we obtain some bounds on the dynamics allowing us to prove the convergence in law of the Markov chain on the system to the solution of a stochastic differential equation, via the infinitesimal generators.     

Degenerate relative equilibria, curvature of the momentum map, and homoclinic bifurcation

A fundamental class of solutions of symmetric Hamiltonian systems is relative equilibria. In this paper the nonlinear problem near a degenerate relative equilibrium is considered. The degeneracy creates a saddle-center and attendant homoclinic bifurcation in the reduced system transverse to the group orbit. The surprising result is that the curvature of the pullback of the momentum map to the Lie algebra determines the normal form for the homoclinic bifurcation. There is also an induced directional geometric phase in the homoclinic bifurcation. The backbone of the analysis is the use of singularity theory for smooth mappings between manifolds applied to the pullback of the momentum map. The theory is constructive and generalities are given for symmetric Hamiltonian systems on a vector space of dimension (2n+2) with an n-dimensional abelian symmetry group. Examples for n=1,2,3 are presented to illustrate application of the theory.     

Modelling of flexible link manipulators

The purpose of this work is to present a method for symbolic generation of the dynamic equations within the Hamiltonian formalism of flexible robots with open‐chain linkage mechanisms using the so‐called vector‐parameterization of the SO(3) group as it is quite effective for many rotational degrees‐of‐freedom. The both, an exact treatment and approximated model of the differential equations of motion are considered. The authors step on their long experience of using the vector‐parameterization for modelling of rigid body mechanical systems. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

时滞Lagrange系统的Lie对称性与守恒量研究

研究了位形间中含单时滞参数的非保守力学系统的Lie对称性和守恒量。首先,利用含时滞的动力学Hamilton原理,建立了含时滞的非保守系统的分段Lagrange运动方程;其次,利用微分方程容许Lie群理论,得到系统的Lie对称确定方程;然后,根据对称性与守恒量之间的关系,通过构造结构方程,得到含时滞的非保守系统的Lie定理;最后,给出了两个具体的算例说明了方法的应用。结果表明:时滞参数的存在使非保守系统的Lagrange方程呈现分段特性,相应的Lie对称性确定方程的个数应是自由度数目的2倍,这对生成元函数提出了更高的限制,同时,守恒量呈现依赖速度项的分段表达。     

Hamiltonian formulation of generalized quantum dynamics

The Hamiltonian formulation of the usual complex quantum mechanics in the theory of generalized quantum dynamics is discussed. After the total trace Lagrangian, total trace Hamiltonian and two kinds of Poisson brackets are introduced, both the equations of motion of some total trace functionals which are expressed by total trace Poisson brackets and the equations of motion of some operators which are expressed by the without-total-trace Poisson brackets are obtained. Then a set of basic equations of motion of the usual complex quantum mechanics are obtained, which are also expressed by the Poisson brackets and total trace Hamiltonian in the generalized quantum dynamics. The set of equations of motion are consistent with the corresponding Heisenberg equations. Project supported by Prof. T.D. Lee’s NNSC Grant, the National Natural Science Foundation of China, the Foundation of Ph. D. Directing Programme of Chinese University, and the Chinese Academy of Sciences.     

SBA splitting for approximation of invariants in time-dependent Hamiltonian systems

Most physical phenomena are described by time-dependent Hamiltonian systems with qualitative features that should be preserved by numerical integrators used for approximating their dynamics. The initial energy of the system together with the energy added or subtracted by the outside forces, represent a conserved quantity of the motion. For a class of time-dependent Hamiltonian systems [8] this invariant can be defined by means of an auxiliary function whose dynamics has to be integrated simultaneously with the system’s equations. We propose splitting procedures featured by a SB³A property that allows to construct composition methods with a reduced number of determining order equations and to provide the same high accuracy for both the dynamics and the preservation of the invariant quantity.     

Stability analysis for Lotka–Volterra systems based on an algorithm of real root isolation

Based on the theory of monotone flows of solutions of systems of differential equations, the Routh–Hurwitz theorem and a real root isolation algorithm of multivariate polynomials are applied to a class of Lotka–Volterra diffusion systems. An algorithm to determine the location of equilibria and the stability of the nonlinear dynamics systems is implemented in Maple.     

A note on the disturbed equations of motion in the non-holonomic coordinates

Summary The aim of this article is to establish the disturbed equations of motion in the non-holonomic coordinates. The governing differential equations of non disturbed motion are the Hamel-Boltzman equations. The disturbed equations obtained in this article can be used for consideration in holonomic and non-holonomic dynamics systems. Entrata in Redazione 1'8 maggio 1972.