Wave solutions of equations of Timoshenko type for the transverse vibrations of plates with parameters periodic on one coordinate

The hyperbolic system of equations that describes the vibrations of plates inhomogeneous along one rectangular coordinate in the context of the Timoshenko theory is presented in canonical hamiltonian form, assuming the solution is periodic on a second coordinate. In the case of periodic inhomogeneity we study the structure of the solutions of certain wave boundary-value problems for plates of this type using the general properties of periodic hamiltonian systems. Bibliography: 6 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 26, 1996, pp. 105–111.     

Area-Preserving Normalizations for Centers of Planar Hamiltonian Systems

It is well known that a nondegenerate center of an analytic Hamiltonian planar system can be brought to normal form by means of an analytic canonical change of coordinates. This normal form, that we denote by CNF, does not depend on the coordinate transformation. In this paper we give an elementary proof of these facts and we show some interesting applications of the machinery that we develop in order to prove them. For instance, we describe the space of coordinate transformations that bring a Hamiltonian nondegenerate center to its CNF, and we prove that they are all canonical when the center is non-isochronous. We also show that two Hamiltonian systems with a nondegenerate center are canonically conjugated if and only if both centers have the same period function.     

Reparameterization-Invariant Reduction in the Hamiltonian Description of a Relativistic String

We study the time-reparameterization-invariant dynamics of an open relativistic string using the generalized Dirac–Hamilton theory and resolving the constraints of the first kind. The reparameterization-invariant evolution variable is the time coordinate of the string center of mass. Using a transformation that preserves the diffeomorphism group of the generalized Hamiltonian and the Poincaré covariance of the local constraints, we segregate the center-of-mass coordinates from the local degrees of freedom of the string. We identify the time coordinate of the string center of mass and the proper time measured in the string frame of reference using the Levi-Civita–Shanmugadhasan canonical transformation, which transforms the global constraint (the mass shell) in the new momentum such that the Hamiltonian reduction does not require the corresponding gauge condition. Resolving the local constraints, we obtain an equivalent reduced system whose Hamiltonian describes the evolution w.r.t. the proper time of the string center of mass. The Röhrlich quantum relativistic string theory, which includes the Virasoro operators L _n only with n > 0, is used to quantize this system. In our approach, the standard problems that appear in the traditional quantization scheme, including the space–time dimension D = 26 and the tachyon emergence, arise only in the case of a massless string, M ² = 0.     

Countably Periodic Gibbs Measures of the Ising Model on the Cayley Tree

We diagonalize the metric Hamiltonian and evaluate the energy spectrum of the corresponding quasiparticles for a scalar field coupled to a curvature in the case of an N-dimensional homogeneous isotropic space. The energy spectrum for the quasiparticles corresponding to the diagonal form of the canonical Hamiltonian is also evaluated. We construct a modified energy–momentum tensor with the following properties: for the conformal scalar field, it coincides with the metric energy–momentum tensor; the energies of the particles corresponding to its diagonal form are equal to the oscillator frequency; and the number of such particles created in a nonstationary metric is finite. We show that the Hamiltonian defined by the modified energy–momentum tensor can be obtained as the canonical Hamiltonian under a certain choice of variables.     

Adiabatic Invariance and Passage Through Resonance for Nearly Periodic Hamiltonian Systems

This is a continuation of previous work on passage through resonance to nearly periodic Hamiltonian systems. We review the classical technique for calculating adiabatic invariants and exhibit the occurrence of zero divisors in the results as a certain critical term evolves slowly through a resonance condition. We then isolate the coordinate associated with the singularity and remove the remaining coordinate from the Hamiltonian to any desired order by successive canonical transformations. This is a variant of the von Zeipel procedure used extensively in celestial mechanics. The momentum conjugate to the cyclic coordinate is an adiabatic invariant, and the reduced Hamiltonian is then solved by constructing and matching three multiple variable expansions which describe the solution before, during, and after resonance passage.     

Three‐dimensional internal gravity‐capillary waves in finite depth

We consider three‐dimensional inviscid‐irrotational flow in a two‐layer fluid under the effects of gravity and surface tension, where the upper fluid is bounded above by a rigid lid and the lower fluid is bounded below by a flat bottom. We use a spatial dynamics approach and formulate the steady Euler equations as an infinite‐dimensional Hamiltonian system, where an unbounded spatial direction x is considered as a time‐like coordinate. In addition, we consider wave motions that are periodic in another direction z. By analyzing the dispersion relation, we detect several bifurcation scenarios, two of which we study further: a type of 00(is)(iκ₀) resonance and a Hamiltonian Hopf bifurcation. The bifurcations are investigated by performing a center‐manifold reduction, which yields a finite‐dimensional Hamiltonian system. For this finite‐dimensional system, we establish the existence of periodic and homoclinic orbits, which correspond to, respectively, doubly periodic travelling waves and oblique travelling waves with a dark or bright solitary wave profile in the x direction. The former are obtained using a variational Lyapunov‐Schmidt reduction and the latter by first applying a normal form transformation and then studying the resulting canonical system of equations.     

Nonlinear vibrations of circular plates with notches. Method of <Emphasis Type="Italic">R</Emphasis>-functions

We investigate vibrations of geometrically nonlinear circular plates with two notches. For the determination of natural frequencies of vibrations, the method of R-functions is used. Nonlinear vibrations of a plate are expanded in eigenmodes of linear vibrations containing R-functions. As a result of using the Bubnov–Galerkin method, we obtain a dynamic system with three degrees of freedom, which is investigated by the method of multiple scales.     

Systems of <Emphasis Type="Italic">sl</Emphasis>(2, ℂ) tops as two-particle systems

We show that Euler-Arnold tops on the algebra sl(2, ℂ) are equivalent to a two-particle system of Calogero type. We show that an arbitrary quadratic Hamiltonian of an sl(2, ℂ) top can be reduced to one of the three canonical Hamiltonians using the automorphism group of the algebra. For each canonical Hamiltonian, we obtain the corresponding two-particle system and write the bosonization formulas for the coadjoint orbits explicitly. We discuss the relation of the obtained formulas to nondynamical Antonov-Zabrodin-Hasegawa R-matrices for Calogero-Sutherland systems. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 157, No. 1, pp. 8–21, October, 2008.     

Canonical formalism for parameter-invariant integrals in the Calculus of Variations whose Lagrange functions involve second order derivatives

Summary In a recent paper [4] a general theory of parameter-invariant integrals in the Calculus of Variations whose Lagrangians involve higher derivatives was developed, and in particular a certain canonical formalism for such problems was discussed. From the point of view of applications it was found that this formalism proved inadequate inas-much as the suggested Hamiltonian function did not depend explicitly on the first derivatives of the positional coordinates. In the present note an alternative Hamiltonian function is defined, which gives rise to a new canonical formalism. The latter is less complicated than the formalism suggested in [4] and is more readily applicable to special problems. A brief discussion of the resulting Hamilton-Jacobi theory is given, and in conclusion the method is illustrated explicitly by means of an example of fairly general character.     

The classical analog of intertwining operators in the relativistic Hamiltonian mechanics of a system of particles

In the context of nonquantum Hamiltonian formalism of the relativistic theory of direct interaction we construct a canonical transformation of the collective variables of center of mass type which transforms the canonical generators of the Poincaré algebra in one form of dynamics into the corresponding generators in another form of dynamics.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 32, 1990, pp. 62–65.     

On the integrability and perturbation of three-dimensional fluid flows with symmetry

Summary The purpose of this paper is to develop analytical methods for studyingparticle paths in a class of three-dimensional incompressible fluid flows. In this paper we study three-dimensionalvolume preserving vector fields that are invariant under the action of a one-parameter symmetry group whose infinitesimal generator is autonomous and volume-preserving. We show that there exists a coordinate system in which the vector field assumes a simple form. In particular, the evolution of two of the coordinates is governed by a time-dependent, one-degree-of-freedom Hamiltonian system with the evolution of the remaining coordinate being governed by a first-order differential equation that depends only on the other two coordinates and time. The new coordinates depend only on the symmetry group of the vector field. Therefore they arefield-independent. The coordinate transformation is constructive. If the vector field is time-independent, then it possesses an integral of motion. Moreover, we show that the system can be further reduced toaction-angle-angle coordinates. These are analogous to the familiar action-angle variables from Hamiltonian mechanics and are quite useful for perturbative studies of the class of systems we consider. In fact, we show how our coordinate transformation puts us in a position to apply recent extensions of the Kolmogorov-Arnold-Moser (KAM) theorem for three-dimensional, volume-preserving maps as well as three-dimensional versions of Melnikov's method. We discuss the integrability of the class of flows considered, and draw an analogy with Clebsch variables in fluid mechanics.     

The Possibility of Forming Coupled Pairs in the Periodic Anderson Model

We investigate the generalized periodic Anderson model describing two groups of strongly correlated (d- and f-) electrons with local hybridization of states and d-electron hopping between lattice sites from the standpoint of the possible appearance of coupled electron pairs in it. The atomic limit of this model admits an exact solution based on the canonical transformation method. The renormalized energy spectrum of the local model is divided into low- and high-energy parts separated by an interval of the order of the Coulomb electron-repulsion energy. The projection of the Hamiltonian on the states in the low-energy part of the spectrum leads to pair-interaction terms appearing for electrons belonging to d- and f-orbitals and to their possible tunneling between these orbitals. In this case, the terms in the Hamiltonian that are due to ion energies and electron hopping are strongly correlated and can be realized only between states that are not twice occupied. The resulting Hamiltonian no longer involves strong couplings, which are suppressed by quantum fluctuations of state hybridization. After linearizing this Hamiltonian in the mean-field approximation, we find the quasiparticle energy spectrum and outline a method for attaining self-consistency of the order parameters of the superconducting phase. For simplicity, we perform all calculations for a symmetric Anderson model in which the energies of twice occupied d- and f-orbitals are assumed to be the same.     

Homogeneous Stäckel-type systems

A class of Hamiltonian dynamic systems integrated by the variable separation method is considered. The integration for this class is the inversion of an Abel mapping on hyperelliptic curves. We prove that the derivative of the Abel mapping is the Stäckel matrix, which determines a diagonal Riemannian metric and curvilinear orthogonal coordinate systems in a flat space. Lax representations with the spectral parameter are constructed. The corresponding classicalr-matrices are dynamic. It is shown how the class of pointwise canonical transformations can be naturally generalized using the Abel integral reduction theory.     

Construction of Canonical Difference Schemes for Hamiltonian Formalism via Generating Functions

This paper discusses the relationship between canonical maps and generating functions and gives the general Hamilton-Jacobi theory for time-independent Hamiltonian systems. Based on this theory, the general method — the generating function method — of the construction of difference schemes for Hamiltonian systems is considered. The transition of such difference schemes from one time-step to the next is canonical. So they are called the canonical difference schemes. The well known Euler centered scheme is a canonical difference scheme. Its higher order canonical generalisations and other families of canonical difference schemes are given. The construction method proposed in the paper is also applicable to time-dependent Hamiltonian systems.     

Normal Form for Families of Hamiltonian Systems

We consider perturbations of integrable Hamiltonian systems in the neighborhood of normally parabolic invariant tori. Using the techniques of KAM-theory we prove that there exists a canonical transformation that puts the Hamiltonian in normal form up to a remainder of weighted order 2d ＋ 1. And some dynamical consequences are obtained.     

Internal gravity–capillary solitary waves in finite depth

We consider a two‐dimensional inviscid irrotational flow in a two layer fluid under the effects of gravity and interfacial tension. The upper fluid is bounded above by a rigid lid, and the lower fluid is bounded below by a rigid bottom. We use a spatial dynamics approach and formulate the steady Euler equations as a Hamiltonian system, where we consider the unbounded horizontal coordinate x as a time‐like coordinate. The linearization of the Hamiltonian system is studied, and bifurcation curves in the (β,α)‐plane are obtained, where α and β are two parameters. The curves depend on two additional parameters ρ and h, where ρ is the ratio of the densities and h is the ratio of the fluid depths. However, the bifurcation diagram is found to be qualitatively the same as for surface waves. In particular, we find that a Hamiltonian‐Hopf bifurcation, Hamiltonian real 1:1 resonance, and a Hamiltonian 0²‐resonance occur for certain values of (β,α). Of particular interest are solitary wave solutions of the Euler equations. Such solutions correspond to homoclinic solutions of the Hamiltonian system. We investigate the parameter regimes where the Hamiltonian‐Hopf bifurcation and the Hamiltonian real 1:1 resonance occur. In both these cases, we perform a center manifold reduction of the Hamiltonian system and show that homoclinic solutions of the reduced system exist. In contrast to the case of surface waves, we find parameter values ρ and h for which the leading order nonlinear term in the reduced system vanishes. We make a detailed analysis of this phenomenon in the case of the real 1:1 resonance. We also briefly consider the Hamiltonian 0²‐resonance and recover the results found by Kirrmann. Copyright © 2016 John Wiley & Sons, Ltd.     

Various methods of determining the natural frequencies and damping of composite cantilever plates. 3. The Ritz method

The Ritz method was used to determine the frequencies and forms of free vibrations of rectangular cantilever plates made of anisotropic laminated composites. Orthogonal Jacobi and Legendre polynomials were used as coordinate functions. The results of the calculations are in good agreement with the published experimental and calculated data of other authors for plates made of boron and carbon fiber reinforced plastics with different angles of reinforcement of unidirectional layers and different sequence of placing the layers, and also of isotropic plates. The dissipative characteristics in vibrations were determined on the basis of the concept of complex moduli. The solution of the frequency equation with complex coefficients yields a complex frequency; the loss factors are determined from the ratio of the imaginary component of the complex frequency to the real component. For plates of unidirectionally reinforced carbon fiber plastic with different relative length a detailed analysis of the influence of the angle of reinforcement on the interaction and frequency transformation and on the loss factor was carried out. The article shows that the loss factor of a plate depends substantially on the type of vibration mode: bending or torsional. It also examines the asymptotics of the loss factors of plates when their length is increased, and it notes that the binomial model of deformation leads to a noticeable error in the calculation of the loss factor of long plates when the angle of reinforcement lies in the range 20°<<70°.For Communication 2, see [1].Institute of Engineering Science of the Russian Academy of Sciences, St. Petersburg, Russia. St. Petersburg State University, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 33, No. 2, pp. 215–225, March–April, 1997.     

一个二流体系统中非线性水波的Hamilton描述

讨论了一个二流体系统中非线性水波的Hamilton描述,该系统由水平固壁之上的两层常密度不可压无粘流体组成,上表面为自由面.文中将速度势函数展开成垂向坐标的幂级数,在浅水长波的假定下,取下层流体的“动厚度”与上层流体的“折合动厚度”为广义位移、界面上和自由面上的速度势为广义动量,根据Hamilton原理并运用Legendre变换导出该系统的Hamilton正则方程,从而将单层流体情形的结果推广到分层流体的情形.