Dynamic analysis of a rectangular plate subjected to multiple forces moving along a circular path

A multi-degree-of-freedom (m.d.o.f.) system excited by a rough moving surface has been developed to study friction-induced oscillations. The normal degrees of freedom allow for oscillatory normal forces, while the normal-tangential coupling of friction produces parametric excitation in the slipping equations of motion. After a modal change of variables, first order averaging has been used to produce a set of autonomous equations of motion. Eigenvalue analysis of the averaged equations has produced stability predictions for the steady sliding position. Numerical integration of the original system of equations has verified the existence of locally unstable oscillations for a system excited by a rough surface input. The combination of velocity-dependent friction and a harmonically varying normal force have been shown to produce large-amplitude oscillations, in some cases leading to stick-slip responses.     

Estimation of the number of degrees of freedom from chaotic time series

A method for the chaotic time series analysis is proposed. It allows one to determine the number of degrees of freedom involved in oscillations from a single observable. The method has been verified for some known stochastic models.     

Photoconductivity of a two-dimensional electron system with spin-orbit coupling in AC magnetic field

The response of an electron system to a dc probe field is analyzed in the case when an initial deviation of conduction-electron spin degrees of freedom from equilibrium in a microwave magnetic field induces combined resonance transitions in the electron system. It is shown that a perturbation of spin degrees of freedom is converted into kinetic energy and modifies transport coefficients, leading to oscillations of diagonal components of the conductivity tensor.     

The heterotic string and the geometry of the supersymmetric Einstein-Yang-Mills background

It is shown that the Wess-Zumino term of a heterotic superstring in aN = 1,D = 10 supergravity background can be given in terms of the torsion of the background in both versions (IA and IB) of theN = 1,D = 10 supergravity theory. The gauge degrees of freedom of the background are included according to the ideas of Kaluza and Klein. Explicit expressions for the vielbein, connection, torsion, and curvature of a space with 506 bosonic and 16 fermionic coordinates are given.     

Transition between synchronization and chaos in doped GaAs/AlAs superlattices

The transition between chaotic and periodic regimes in spontaneous current oscillations of weakly coupled, doped GaAs/AlAs superlattices has been observed by varying the external d.c. bias. The chaotic current oscillations are observed for voltage ranges, which exhibit a large negative differential conductance in the time-averaged I–V characteristic. Since this system can be described by a spatially distributed, non-linear system with many degrees of freedom, the coupling between the degrees of freedom in the chaotic windows is repulsive, while in the periodic windows it is attractive.     

Effect of magnetoelastic interaction on the thermodynamics of ferromagnets: Simulation by the method of spin-lattice dynamics

Thermodynamic properties of systems with coupled magnetic and lattice degrees of freedom are analyzed by the numerical spin-lattice dynamics (SLD) method. A scheme of numerical integration is developed for SLD equations in a thermostat, that follows the earlier formulated approaches and is modified to describe systems with realistic interatomic interactions. The method proposed allows one to calculate the spectral density of oscillations, heat capacity, magnetization, and thermal expansion coefficient within a single scheme. It is established that, due to short-range magnetic order, the interplay between magnetic and lattice degrees of freedom contributes to the thermodynamic properties of the system even in the paramagnetic state. It is shown that there exist two mechanisms how the spin-lattice interaction influences the thermodynamic properties: static and dynamic mechanisms; the first is determined by its contribution to the thermal expansion of the lattice, and the second, by the dynamic interaction between magnetic moments and crystal lattice vibrations.     

The dynamical structure of higher dimensional Chern-Simons theory

Higher dimensional Chern-Simons theorìes, even though constructed along the same topological pattern as in 2 + 1 dimensions, have been shown recently to have generically a non-vanishing number of degrees of freedom. In this paper, we carry out the complete Dirac Hamiltonian analysis (separation of first and second class constraints and calculation of the Dirac bracket) for a group G × U(1). We also study the algebra of surface charges that arise in the presence of boundaries and show that it is isomorphic to the WZW₄ discussed in the literature. Some applications are then considered. It is shown, in particular, that Chern-Simons gravity in dimensions greater than or equal to five has a propagating torsion.     

Arnol’d diffusion in a system with 2.5 degrees of freedom: Classical and quantum mechanical approaches

Arnol’d diffusion, a universal phenomenon in nonlinear dynamics, is analyzed for a model system with 2.5 degrees of freedom. Only the three primary order resonances are taken into account, and the results obtained by using classical and quantum mechanical approaches are compared. It is shown that the parameter dependence of the rate of quantum Arnol’d diffusion is similar to the classical one, but the quantum diffusion coefficient is smaller by approximately an order of magnitude. It is found that the existence of a threshold with respect to perturbation parameters, pointed out earlier, is not an indispensable feature of quantum Arnol’d diffusion. It is shown that a quantum system with weakly overlapping resonances can exhibit mixed dynamics that has no classical counterpart (diffusion along a resonance superimposed by oscillations across the overlapped resonances).     

Anharmonic interactions of elastic and orientational waves in one-dimensional crystals

Planar oscillations of a chain of dumbbell-shaped particles possessing three degrees of freedom are studied. This system models the dynamics of quasi-one-dimensional crystals consisting of elongated anisotropic molecules. A system of nonlinear differential equations describing the anharmonic interaction of the elastic and orientational waves in the lattice, corresponding to different degrees of freedom of the particles, is constructed assuming a cubic interparticle interaction potential. It is shown that in the low-frequency approximation the system obtained is equivalent to the equations of the moment theory of elasticity, widely employed for describing nonlinear and dispersion properties of layered crystals and phase transformations in alloys. Some types of three-wave collinear interactions are investigated, suggesting the possibility of exciting orientational waves in organic crystals because of their nonlinear interaction with acoustic waves. Fiz. Tverd. Tela (St. Petersburg) 39, 137–144 (January 1997)     

On teleportation in a system of identical particles

The teleportation of an unknown polarization state of one of the photons in a system of identical particles has been considered. It has been shown that the spatial degrees of freedom, which are various directions of the momentum of three photons, are of significant importance for teleportation in the system of identical particles. The inclusion of the spatial degrees of freedom increases the dimension of the space of single-particle states. In view of this increase, a four-dimensional subspace of two-particle states, which is similar to the space of states spanned on the Bell states in the system of two distinguishable qubits, can be separated in the experimental configuration.     

A geometrical theory of spin motion

A discussion of the fundamental interrelation of geometry and physical laws with Lie groups leads to a reformulation and heuristic modification of the principle of inertia and the principle of equivalence, which is based on the simple de Sitter group instead of the Poincaré group. The resulting law of motion allows a unified formulation for structureless and spinning test particles. A metrical theory of gravitation is constructed with the modified principle, which is structured after the geometry of the manifold of the de Sitter group. The theory is equivalent to a particular Kaluza-Klein theory in ten dimensions with the Lorentz group as gauge group. A restricted version of this theory excludes torsion. It is shown by a reformulation of the energy momentum complex that this version is equivalent to general relativity with a cosmologic term quadratic in the curvature tensor and in which the existence of spinning particle fields is inherent from first principles. The equations of the general theory with torsion are presented and it is shown in a special case how the boundary conditions for the torsion degree of freedom have to be chosen such as to treat orbital and spin angular momenta on an equal footing. The possibility of verification of the resulting anomalous spin-spin interaction is mentioned and a model imposed by the group topology ofSO(3,2) is outlined in which the unexplained discrepancy between the magnitude of the discrete valued coupling constants and the gravitational constant in Kaluza-Klein theories is resolved by the identification of identical fermions as one orbit. The mathematical structure can be adapted to larger groups to include other degrees of freedom.     

Linear Transformation of the Polarization Modes in Coiled Optical Spun-Fibers with Strong Unperturbed Linear Birefringence. I. Nonresonant Transformation

A linear transformation of orthogonal polarization modes in coiled optical spun-fibers with strong unperturbed linear birefringence, which causes the emergence of the dependences of the integrated elliptical birefringence and the ellipticity and azimuth of the major axis of the ellipse, as well as the polarization state of radiation (PSR), on the length of optical fiber has been considered. Optical spun-fibers are subjected to a strong mechanical twisting, which is frozen into the structure of the optical fiber upon cooling, in the process of being drawn out from the workpiece. Since the values of the local polarization parameters of coiled spunwaveguides vary according to a rather complex law, the calculations were carried out by numerical modeling of the parameters of the Jones matrices. Since the rotation speed of the axes of the birefringence is constant on a relatively short segment of a coiled optical spun-fiber in the accompanying torsion (helical) coordinate system, the so-called “Ginzburg helical polarization modes” (GHPMs)—two mutually orthogonal ellipses with the opposite directions of traversal, the axis of which rotate relative to the fixed coordinate system uniformly and unidirectionally—are approximately the local normal polarization modes of such optical fiber. It has been shown that, despite the fact that the unperturbed linear birefringence of the spun-fibers significantly exceeds the linear birefringence, which is caused by the winding on a coil, the integral birefringence of an extended segment of such a fiber coincides in order of magnitude with the linear birefringence, which is caused by the winding on the coil, and the integral polarization modes tend asymptotically to circular ones. It has been also shown that the values of the circular birefringence of twisted single-mode fibers, which were calculated in a nonrotating and torsion helical coordinate systems, differ significantly. It has been shown that the polarization phenomena occur in the process of linear transformation of local polarization modes, which lead to small quasi-harmonic oscillations of the birefringence integral parameters of the optical spun-fibers, which depend on their length, and the period of these oscillations is approximately equal to half of the effective period of polarization beating.     

Quantum Mechanics of Systems with a Nonenumerable Number of Degrees of Freedom

Is has been shown that the quantum theory of systems with an infinite nonenumerable number of degrees of freedom is incompatible with the assumption about the continuity of the space of indices of degrees of freedom: in this case, a discrete topology on the space of indices is required. This is illustrated by the example of field systems.     

Interaction Dynamics of the External and Internal Degrees of Freedom of an Atom in the Resonant Field of a Standing Light Wave

The theory of the dynamic interaction of the external (translational) and internal (electronic) degrees of freedom of a twolevel atom in the field of a standing light wave in a perfect cavity of the Fabry–Perot type was developed. The theory describes the energy exchange between three subsystems, namely, translational, electronic, and field subsystems, as opposed to the theories of the parametric interaction (in the approximations of Raman–Nath and/or large resonance detuning) and of the atomic motion in free space. In the semiclassical approximation, the corresponding Heisenberg equations of motion were shown to form a closed Hamiltonian dynamic system with two degrees of freedom, namely, translational and collective electron–field degrees of freedom. This system is integrated in terms of the elliptic Jacobian functions in the resonance limit. The solutions obtained describe the effects of trapping of an atom in the periodic potential of the standing light wave, and its cooling and heating, as well as the effect of the dynamic Rabi oscillations. The latter is caused by the interaction of the internal and external atomic degrees of freedom through the radiation field.     

高自由度体系微振动本征模式的求解方法

介绍两种求解高自由度体系微振动本征模式的方法．第一种方法：利用对称性经过物理分析寻找本征模式；第二种方法：利用对称性降低体系的自由度，然后用求解本征模式的一般方法解之．     

Quantum Nonlinear Oscillator with Two Degrees of Freedom in a Laser Field

The semiclassical dynamics of a quantum nonlinear oscillator with two degrees of freedom and anharmonicity of the fourth order in a periodic laser field is studied both analytically and numerically. In the absence of external excitation and dissipation, the equations of motion for the mean values of the coordinate and momentum operators of both degrees of freedom reduce to the equation of a onedimensional nonlinear pendulum. The general solution of this equation is written in terms of the Jacobian elliptic functions. As can be expected, the energy of the free oscillator is redistributed periodically between degrees of freedom. The periodic excitation of the nonlinear oscillator may substantially change its motion pattern. Using as an example an oscillator with two coupled vibrational degrees of freedom, it is numerically shown that the amount of laser photons absorbed depending on the parameter values and initial conditions may vary with time in a rather complex manner, including chaotic oscillations. A nonlinear oscillator is capable of manifesting bistable behavior with allowance for dissipation. The analytical condition for the origination of bistability is found. Examples of the bistable dependence of the number of quanta in the oscillator vibrational mode on the level of laser excitation are presented.     

A systematic approach to the calculation of tricritical temperature with application to the nematic-smecticA phase transition

Using the Landau expansion equations have been found that allow for systematic calculations of tricritical temperature for a given molecular model. The theory has been applied to nematic-smecticA (NA) phase transition of liquid crystals. It has been shown exactly that the NA tricritical temperature depends only on the couplings between the two lowest order translational order parameters and the orientational degrees of freedom. Numerical results of Mayer and Lubensky for the NA tricritical point have been derived exactly. Also a stability condition of the obtained solution has been discussed.Alexander von Humboldt Foundation Fellow, 1985–1986     

Vibration and waves in a cylinder with screw anisotropy

On the basis of the three-dimensional theory of elasticity, we consider the features of propagation of harmonic waves in a hollow cylinder with screw anisotropy. The main attention is focused on studying axisymmetrical oscillations. To describe low-frequency long-wavelength longitudinal-torsional oscillations by perturbation theory methods, we construct an applied theory and evaluate the area of its applicability. To analyze high-frequency oscillations, a numerical method of determining the critical frequencies and dispersion curves is developed and implemented. It is shown that in the axisymmetrical case, screw anisotropy generates a relation between longitudinal and torsion oscillations, which is mathematically described by the amplitude coefficients of homogeneous waves.     

On non-linear oscillations with slowly varying system parameters

The paper deals with an approximate analysis of non-linear oscillation problems with slowly varying system parameters. From the differential equations for amplitude and phase, set up by the method of variation of parameters, the approximate solutions are obtained by using the generalized averaging method of Sinha and Srinivasan based on ultraspherical polynomial expansions. The Bogoliubov-Mitropolsky results are given by a particular set of these polynomials. Problems of a single degree of freedom system as well as monofrequency oscillations in systems with multiple degrees of freedom are considered. The approach has been illustrated by an example and the results are compared with the numerical solutions. A close agreement is found.     

Torsion Degrees of Freedom in the Regge Calculus as Dislocations on the Simplicial Lattice

Using the notion of a general conical defect, the Regge Calculus is generalized by allowing for dislocations on the simplicial lattice in addition to the usual disclinations. Since disclinations and dislocations correspond to curvature and torsion singularities, respectively, the method we propose provides a natural way of discretizing gravitational theories with torsion degrees of freedom like the Einstein-Cartan theory. A discrete version of the Einstein-Cartan action is given and field equations are derived, demanding stationarity of the action with respect to the discrete variables of the theory.