Coisotropic quasi-periodic motions near the relative equilibrium of a Hamiltonian system

Abstract

We consider the Hamiltonian system which is invariant under locally Hamiltonian (non-Poissonian) action of torus. We show that when a certain set of conditions is satisfied the majority of motions in a sufficiently small neighbourhood of system’s relative equilibrium are quasi-periodic and cover coisotropic invariant tori.     

Note on coupled motions of vortices and rigid bodies

The equations of motion for a cylinder of radius a, in a perfect planar fluid, with one point vortex, have a sui generis hamiltonian structure, as it is exhibited in the limit case a→0. Reduction is accomplished by the method of Marsden and Weinstein.     

Alternative formulation in the constrained Hamiltonian system of a rigid rotator

An alternative form of the constraint characterizing the motion of a rigid rotator is introduced and is shown to lead to the same constrained Hamiltonian system as the one constructed with the conventional constraintx ²=constant. In this new formulation, the canonical pairs of phase-space variables can easily be found, and the relationx ²=constant appears through an equation of motion rather than through a constraint. Application toO(N) nonlinear sigma model is also discussed.     

A relativistic generalisation of rigid motions

A weaker substitute for the too restrictive class of Born-rigid motions is proposed, which we call radar-holonomic motions. The definition is expressed as a set of differential equations. Integrability conditions and Cauchy problem are studied. We finally obtain an example of a radar-holonomic congruence containing a given worldline with a given value of the rotation on this line.     

Regular and chaotic motions in applied dynamics of a rigid body

Periodic and regular motions, having a predictable functioning mode, play an important role in many problems of dynamics. The achievements of mathematics and mechanics (beginning with Poincare) have made it possible to establish that such motion modes, generally speaking, are local and form "islands" of regularity in a "chaotic sea" of essentially unpredictable trajectories. The development of computer techniques together with theoretical investigations makes it possible to study the global structure of the phase space of many problems having applied significance. A review of a number of such problems, considered by the authors in the past four or five years, is given in this paper. These include orientation and rotation problems of artificial and natural celestial bodies and the problem of controlling the motion of a locomotion robot. The structure of phase space is investigated for these problems. The phase trajectories of the motion are constructed by a numerical implementation of the Poincare point map method. Distinctions are made between regular (or resonance), quasiregular (or conditionally periodic), and chaotic trajectories. The evolution of the phase picture as the parameters are varied is investigated. A large number of "phase portraits" gives a notion of the arrangement and size of the stability islands in the "sea" of chaotic motions, about the appearance and disappearance of these islands as the parameters are varied, etc. (c) 1996 American Institute of Physics.     

Plastic deformation and quasi-periodic vibrations in a tribological system

A one-dimensional macroscopic model is used to analyze the plastic deformation of materials without coating and with a plastic hardening coating or a plastic nonhardening coating at friction. The calculations show that mechanical vibrations can be excited in a tribological system and that their frequency decreases sharply when going from elastic to plastic deformation. One of the causes of the development of plastic deformation in the surface layer and in the sublayer of the material under a hard coating is found to be a decrease in the elastic properties of the material because of frictional heating. An intense plastic shear in the material under the hard coating can cause its failure due to incompatible strains of the coating and the base.     

Integrability of two interactingN-dimensional rigid bodies

A new class of integrable Euler equations on the Lie algebra so(2n) describing twon-dimensional interacting rigid bodies is found. A Lax representation of equations of motion which depends on a spectral parameter is given and complete integrability is proved. The double hamiltonian structure and the Lax representation of the general flow is discussed.On leave of absence from the Institute for Theoretical Physics of Warsaw University, ul. Hoza 69, PL-00-681 Warsaw, Poland     

Trotter-formula for the time evolution of quasi-periodic quantum system

In this paper we investigate the time evolution operator for a class of Schrödinger operators for which a classical Lagrangian is not defined.We evaluate explicitly the strong limit of the Trotter approximation series.     

Absence of radiationless motions of relativistically rigid classical electron

Radiationless motion of a charge distribution is reviewed, and the necessary condition conjectured by Goedecke is proved. Then it is shown that a nonrotating, uniformly charged, spherical shell (as seen from its own rest frame) which moves in a relativistically invariant fashion does not have any bounded radiationless motions, unlike its nonrelativistic counterpart.     

A connection theory for a nonlinear differential constrained system

An Ehresmann connection on a constrained state bundle defined by nonlinear differential constraints is constructed for nonlinear nonholonomic systems. A set of differential constraints is integrable if and only if the curvature of the Ehresmann connection vanishes. Based on a geometric interpretation of d－δ commutation relations in constrained dynamics given in this paper, the complete integrability conditions for the differential constraints are proven to be equivalent to the three requirements upon the conditional variation in mechanics: (1) the variations belong to the constrained manifold; (2) the time derivative commutes with variational operator; (3) the variations satisfy the Chetaev's conditions.     

Development of a quasi-periodic undulator for the HLS

China's first quasi-periodic undulator (QPU) has been developed for the Hefei Light Source (HLS). It uses a magnetic configuration with varied thicknesses of NdFeB blocks, which is based on the QPU of European Synchrotron Radiation Facility (ESRF). The depression of 3^rd harmonic radiation is significantly improved over the ESRF QPU, as deduced from the measured magnetic fields. A method of configuring shims of different geometries and sizes, based on a symmetric principle to correct multi-pole field integrals, is demonstrated.     

Expressions for forces and torques in molecular simulations using rigid bodies

Expressions for intermolecular forces and torques, derived from pair potentials between rigid non-spherical units, are presented. The aim is to give compact and clear expressions, which are easily generalized, and which minimize the risk of error in writing molecular dynamics simulation programs. It is anticipated that these expressions will be useful in the simulation of liquid crystalline systems, and in coarse-grained modelling of macromolecules.     

Deformations of Coisotropic Submanifolds for Fibrewise Entire Poisson Structures

We show that deformations of a coisotropic submanifold inside a fibrewise entire Poisson manifold are controlled by the L _∞-algebra introduced by Oh–Park (for symplectic manifolds) and Cattaneo–Felder. In the symplectic case, we recover results previously obtained by Oh–Park. Moreover we consider the extended deformation problem and prove its obstructedness.     

Stability of noncircular and infinite motions of test bodies in general relativity

The stability of finite noncircular and infinite motions of nonspinning and spinning test bodies in the Schwarzschild field has been investigated. It is shown that these motions are unstable by Lyapunov relative to variables . The conditions of the stability in a finite interval of time for these motions have been found. The time interval of the motion stability has been numerically estimated.     

Stability analysis of a simple rheonomic nonholonomic constrained system

It is a difficult problem to study the stability of the rheonomic and nonholonomic mechanical systems. Especially it is difficult to construct the Lyapunov function directly from the differential equation. But the gradient system is exactly suitable to study the stability of a dynamical system with the aid of the Lyapunov function. The stability of the solution for a simple rheonomic nonholonomic constrained system is studied in this paper. Firstly, the differential equations of motion of the system are established. Secondly, a problem in which the generalized forces are exerted on the system such that the solution is stable is proposed. Finally, the stable solutions of the rheonomic nonholonomic system can be constructed by using the gradient systems.