Classical Mathematical Physics Research Articles

Galilean relativity and the path integral formalism in quantum mechanics

Closed systems in Newtonian mechanics obey the principle of Galilean relativity. However, the usual Lagrangian for Newtonian mechanics, formed from the difference of kinetic and potential energies, is not invariant under the full group of Galilean transformations. In quantum mechanics, Galilean boosts require a non-trivial transformation rule for the wave function and a concomitant “projective representation” of the Galilean symmetry group. Using Feynman's path integral formalism, this latter result can be shown to be equivalent to the non-invariance of the Lagrangian. Thus, using path integral methods, the representation of certain symmetry groups in quantum mechanics can be simply understood in terms of the transformation properties of the classical Lagrangian and conversely. The main results reported here should be accessible to students and teachers of physics—particularly classical mechanics, quantum mechanics, and mathematical physics—at the advanced undergraduate and beginning graduate levels, providing a useful exposition for those wanting to explore topics such as the path integral formalism for quantum mechanics, relativity principles, Lagrangian mechanics, and representations of symmetries in classical and quantum mechanics.

Wave patterns and dynamical properties of optical propagation by a higher order nonlinear Schrödinger equation

It is of great significance to explore the physical properties of optical propagation, thus the scholars are keen on the physical essence represented by classical mathematical physics equations. In this paper, a higher order nonlinear Schrödinger equation describing the behavior of polarization mode in optical fibers is firstly analyzed qualitatively, and the existence of periodic and soliton solutions is proved by using bifurcation method. Secondly, all chirped wave patterns with special form for the equation are obtained by using the complete discrimination system for polynomial method and direct integral method, the chirped rational functions and some elliptic functions wave patterns are initially found. As a result, the parameter stability of these patterns is given for the first time, which shows the variant of patterns as the parameters change, and the graphs of several typical patterns are drawn.

A simple calculation model for transport of visitors on a closed circuit

It is examined a very simple, geometrically closed configuration concerning transport systems. The problem analyzed is a multi-objective optimization problem, in which an ever increasing set of visitors is urged to visit and move from a given set of sites on a closed path. An essentially kinematic approach is developed and the performance evaluation is obtained by means of a combinations of variables. It is also examined a particular realization in which one makes use of C_0-functions instead of discrete variables, in the spirit of classical mathematical physics. Some features of the model show relevant differences with others concerned with traffic and transport problems, for the presence of Diophantine integral evolution equations in place of statistical and/or numerical complex evaluation methods. This work is the first part of a more thorough discussion on dynamical equations in transport systems, including simulations and optimization schemes.

Effects of long-term leakage of shield lining on tunnelling-induced ground consolidation movements

Limited mathematical analytical studies for tunnel leakage-induced surface and subsurface soil movements and tunnel responses incorporated the distribution of excess pore water pressure, in which the concrete lining was often assumed perfectly permeable or impermeable. This mechanical closed-form model will bring calculation errors and inconsistencies in engineering practice. This article introduces a new set of mathematical model for analyzing tunnelling-induced dissipation of excess pore water pressure and ground consolidation settlement considering long-term leakage influence through a semi-permeable lining, in which the soil-lining permeability ratio is used as the relative parameter to measure the influence of lining permeability on consolidation settlement. The interesting point is that the theoretical model conducts coupling with the Terzaghi-Rendulic consolidation theory through solving of the conformal mapping technique in complex variable theory and the separation variable method in classical mathematical physics equation. The accuracy of presented solutions is verified by comparisons with numerical simulation and in-situ measurements of engineering profiles and reasonable agreements are obtained. Furthermore, a sensitivity analysis is implemented on the subsurface consolidation settlement and the distribution of excess pore water pressure, and the results indicate that the soil-lining permeability and the soil Young’s modulus notably affect the tunnel leakage-induced ground responses.

MASS TRANSFER PROCESSES DURING CORROSION OF CEMENT CONCRETE OF ROAD PAVEMENT AND AIRFIELD COVERINGS MODELING

Varieties of corrosion processes under the action of various corrosive environments on the cement concretes for pavement and airfield coatings within the classification by V.M. Moskvin are considered and their distinctive features are described. It is noted in the work that the rate of destruction of cement concretes is determined by the processes of mass transfer and chemical reactions. On this basis, we provide the substantiation of a generalized methodological approach to modeling the processes of mass transfer during liquid corrosion of building materials. For corrosion types I and II, analytical solutions are obtained for specific initial and boundary conditions of the problem.
 Development of mathematical models of concrete corrosion processes is based on the physical models of diffusion of components that are imbedded in the porous structure of concrete and the mathematical apparatus of boundary problems of mass transfer using differential equations in partial derivatives of parabolic type. The works, to which the authors refer, provide the results of the development of mathematical models of the processes of corrosion of concrete of the first type, as well as the results of the practical application of these mathematical models. However, the results obtained cannot withstand scrutiny in terms of methods and approaches of classical mathematical physics.
 In this study, these errors, inaccuracies, inconsistencies and incorrectness of the abovementioned works are eliminated. At the same time, models of corrosion of concretes of the first and second types are taken into account. An undoubtedly positive aspect of the dependences obtained in the study is the possibility of solving the inverse problem, when the actual experimental data using this model make it possible to predict the numerical values of free calcium hydroxide, which, as a result, with a minimum error allows to predict the long-term durability of structures and buildings.

On a Hilbert space framework for evolutionary systems of classical mathematical physics: a survey

We survey a space–time Hilbert space approach to familiar dynamical partial differential equation problems from mathematical physics of the operator normal form $$\left( \partial _{0}{\mathcal {M}}+A\right) U=F,$$ F given. Here, $$\partial _{0}$$ denotes the time-derivative, A is a skew-selfadjoint spatial operator and $${\mathcal {M}}$$ describes the material properties. For sake of simplicity, we focus on the case of $${\mathcal {M}}$$ being such that $$\partial _{0}{\mathcal {M}}=\partial _{0}M_{0}+M_{1}$$ , where $$M_{0},M_{1}$$ are continuous spatial operators, the case of materials with no memory. This special case is shown to be closely related to space–time Friedrichs systems. The framework is illustrated with examples such as the classical acoustic, electromagnetic and elastic wave propagation, as well as the Klein–Gordon, Schrodinger and Dirac equation.

Rays, Waves, and Scattering: Topics in Classical Mathematical Physics

Rays, Waves, and Scattering: Topics in Classical Mathematical Physics

On recent advances in boundary-value problems in the plane

The survey is devoted to recent advances in nonclassical solutions of the main boundary-value problems such as the well-known Dirichlet, Hilbert, Neumann, Poincare, and Riemann problems in the plane. Such solutions are essentially different from the variational solutions of the classical mathematical physics and based on the nonstandard point of view of the geometric function theory with a clear visual sense. The traditional approach of the latter is the meaning of the boundary values of functions in the sense of the so-called angular limits or limits along certain classes of curves terminated at the boundary. This become necessary if we start to consider boundary data that are only measurable, and it is turned out to be useful under the study of problems in the field of mathematical physics as well. Thus, we essentially widen the notion of solutions and, furthermore, obtain spaces of solutions of the infinite dimension for all the given boundary-value problems. The latter concerns the Laplace equation, as well as its counterparts in the potential theory for inhomogeneous and anisotropic media.

Introduction to the special volume ``Mathematics of nonlinear acoustics: New approaches in analysis and modeling''

Over the last 12--15 years, there has been a resurgence of interest in the study of nonlinear acoustic phenomena. Using the tools of both classical mathematical analysis and computational physics, researchers have obtained a wide range of new results, some of which might be described as remarkable. As with almost all trends in science, the reasons for this revival are varied: they range from practical applications (e.g., the need to improve our understanding of high-intensity ultrasound); to the development of numerical schemes which are better at capturing the physics of nonlinear compressible flow; to new acoustic models which lend themselves to study by analytical methods.For more information please click the “Full Text” above.

Toward the Landau–Lifshitz version of the Thomson electrostatics theorem

In the classical textbook (Landau and Lifshitz, 1963), Landau and Lifshtz suggested their version of the famous Thomson variational principle (a.k.a Thomson theorem.) So far, their version has not gained the interest it deserves, either among physicists or among applied mathematicians. Partially, the lack of interest can be explained because of the quality of the suggested proof of the principle. It is considerably lower than the standards accepted in classical electrostatics and mathematical physics. Even more importantly, Landau and Lifshitz did not demostrate the minimum property of the electrostatic energy at equilibrium. In this note, we, first, modify and specify the Landau–Lifshitz formulation of the principle presenting it as the isoperimetric variational problem. Then, for this isoperimetric problem we calculate the first and second variations, and we prove that the first variation vanishes, whereas the second variation appears to be positive.

On a connection between the Maxwell system, the extended Maxwell system, the Dirac operator and gravito‐electromagnetism

The close connection between Maxwell's equations and the Dirac equation is under consideration in a rigorous functional analytical framework. The two systems are linked via the so‐called extended Maxwell system Picard R. A structural observation for linear material laws in classical mathematical physics. Mathematical Methods in the Applied Sciences 2009;32(14):1768–1803, which is here re‐considered in the time‐dependent case as an evolutionary space‐time operator equation. This structural observation is then applied to recover and generalize the equations of gravito‐electromagnetism and to reformulate theMaxwell‐Dirac system as a system of three coupled extended Maxwell systems. This reformulation rests on the observation that what in electrodynamics is commonly known as ‘potential’ is actually a solution to another extended Maxwell system. Copyright © 2014 John Wiley & Sons, Ltd.

The two-dimensional free-space Green’s function and its derivatives in a tangent-normal system

The two-dimensional free-space Green’s function, G(2), and its derivatives, are used extensively in the formulation of scattering and diffraction problems through its presence in single- and double-layer potentials, and their use in integral equations. The vast majority of the results from elementary classical mathematical physics for G(2) is based on Cartesian coordinate-space, either directly as a Hankel function in coordinate-space or through a transform, such as the Weyl transform, also based on Cartesian coordinate-space. However, if the geometry of the problem is not Cartesian, for example in scattering from a rough surface, there are difficulties in using a transform representation for G(2) which depends on Cartesian geometry, as the standard Weyl transform does. Here we formulate transform-space representations using a tangent-normal coordinate system. The result for G(2) is a new Weyl-type tangent-normal transform representation from which the results for the vector derivatives of the single-layer potential, the double-layer potential, and the vector derivatives of the double-layer potential follow quite simply. The latter three results can be expressed in terms of two new spectral functions in tangent-normal space, S1 and S2. The overall results are new representations for G(2) and its derivatives which may be useful in integral equation formulations of scattering problems for non-Cartesian geometries.

Fractal structures in weakly dissipative kam theory and related problems

This paper presents a number of numerical quantities that admit interpretation in terms of classical and quantum mathematical physics.

Homogenization of a class of linear partial differential equations

We present a unified Hilbert space perspective to homogenization of a class of evolutionary equations of mathematical physics. We formulate homogenization in a purely operator-theoretic setting. Using A structural observation for linear material laws in classical mathematical physics by R. Picard (Mathematical Methods in the Applied Sciences 32 (2009), 1768-1803), we discuss constitutive relations as certain elements of the Hardy space H ∞ (E; L(H)) of bounded, analytic and operator- valued functions M : E → L(H), where E C open, H Hilbert space. The core idea is to introduce a certain topology on the set of constitutive relations. Given a convergent sequence of constitutive relations, the behavior of solutions to the respective problems is discussed. We apply the results to the equations of acoustics, thermodynamics, elasticity or coupled systems such as thermo-elasticity. The respective equations may also incorporate memory or delay terms and fractional derivatives. In particular, constitutive relations via differential equations can also be treated.

Exponential stability for linear evolutionary equations

We give an approach to exponential stability within the framework of evolutionary equations due to (R. Picard. A structural observation for linear material laws in classical mathematical physics. Math. Methods Appl. Sci., 32(14):1768-1803, 2009). We derive sufficient conditions for exponential stability in terms of the material law operator which is defined via an analytic and bounded operator-valued function and give an estimate for the expected decay rate. The results are illustrated by three examples: differential-algebraic equations, partial differential equations with finite delay and parabolic integro-differential equations.

Schrödinger and Dirac equations for the hydrogen atom, and Laguerre polynomials

It is usually claimed that the Laguerre polynomials were popularized by Schrodinger when creating wave mechanics; however, we show that he did not immediately identify them in studying the hydrogen atom. In the case of relativistic Dirac equations for an electron in a Coulomb field, Dirac gave only approximations, Gordon and Darwin gave exact solutions, and Pidduck first explicitly and elegantly introduced the Laguerre polynomials, an approach neglected by most modern treatises and articles. That Laguerre polynomials were not very popular before their use in quantum mechanics, probably because they had been little used in classical mathematical physics, is confirmed by the fact that, as we show, they had been rediscovered independently several times during the nineteenth century, in published or unpublished studies of Abel, Murphy, Chebyshev, and Laguerre.

On a comprehensive class of linear material laws in classical mathematical physics

A class of evolutionaryproblems is considered, which covers a number of diverse initial boundaryvalue problems of classical mathematical physics. The claim that thisclass is indeed to a large extent sufficiently general is exemplifiedby some specific models for visco-elastic solids.

A structural observation for linear material laws in classical mathematical physics

AbstractA class of linear material laws is considered, which covers a number of diverse initial boundary‐value problems of classical mathematical physics. The claim that this class is indeed to a large extent sufficiently general is exemplified for a number of specific models from classical physics. Copyright © 2009 John Wiley & Sons, Ltd.

Renormalization-group symmetries for solutions of nonlinear boundary value problems

About 10 years ago, the method of renormalization-group symmetries entered the field of boundary value problems of classical mathematical physics, stemming from the concepts of functional self-similarity and of the Bogoliubov renormalization group treated as a Lie group of continuous transformations. Overwhelmingly dominating practical quantum field theory calculations, the renormalization-group method formed the basis for the discovery of the asymptotic freedom of strong nuclear interactions and underlies the Grand Unification scenario. This paper draws on lectures delivered at the XIII School for Nonlinear Waves, Nizhnii Novgorod, Russia, 1–7 March 2006 [see V F Kovalev, D V Shirkov "Renormalization group symmetry for solutions of boundary value problems" in Nonlinear Waves 2006 (Ed. by A V Gaponov-Grekhov) (N. Novgorod: IAP RAS, 2007) p. 433] to describe the logical framework of a new algorithm based on the modern theory of transformation groups and to present the most interesting results of application of the method to differential and/or integral equation problems and to problems that involve linear functionals of solutions. Examples from nonlinear optics, kinetic theory, and plasma dynamics are given, where new analytic solutions obtained with this algorithm have allowed describing the singularity structure for self-focusing of a laser beam in a nonlinear medium, studying generation of harmonics in weakly inhomogeneous plasma, and investigating the energy spectra of accelerated ions in expanding plasma bunches.

On the difference approximations of overdetermined hyperbolic equations of classical mathematical physics

Issues concerning difference approximations of overdetermined systems of hyperbolic equations are examined. The formulations of extended overdetermined systems are given for hydrodynamics equations, magnetohydrodynamics equations, Maxwell equations, and elasticity equations. Some approaches to the construction of difference schemes are discussed for these systems.