Integral Variational Principles Research Articles

Upper and lower bounds for the speed of fronts of the reaction diffusion equation with Stefan boundary conditions

We establish two integral variational principles for the spreading speed of the one dimensional reaction diffusion equation with Stefan boundary conditions. The first principle is valid for monostable reaction terms and the second principle is valid for arbitrary reaction terms. These principles allow to obtain several upper and lower bounds for the speed. In particular, we construct a generalized Zeldovich–Frank–Kamenetskii type lower bound for the speed and upper bounds in terms of the speed of the standard reaction diffusion problem. We construct asymptotically exact lower bounds previously obtained by perturbation theory.

ВИКЛАДАННЯ ТЕМИ “РІВНЯННЯ ЛАГРАНЖА НА ОСНОВІ ДИФЕРЕНЦІАЛЬНОГО ВАРІАЦІЙНОГО ПРИНЦИПУ Д’АЛАМБЕРА-ЛАГРАНЖА”

While preparing physics and mathematics specialists, and in particular physics teachers, it should be paid attention to the general principles, which in a compact form contain not only all known experimental and theoretical positions, but also allow predicting new discoveries. These principles include integral variational principles that were first formulated in mechanics. Preparing physics teachers’, the principles are studied in the theoretical physics course in its first section – “Classical mechanics”. By studying classical mechanics, unlike the general course “Mechanics”, students encounter many generalized and abstract concepts, the formation of which puts before teachers a lot of methodological problems to be solved. The following methods were used for research: systematic scientific and methodological analysis of textbooks and manuals, articles on the research problem; observation of the educational process; synthesis, comparison and generalization of theoretical positions, discovered in the scientific and educational literature; generalization of own pedagogical experience. The authors of the article summarized the results of the analysis of textbooks and their own experience, and on the basis of this, one of the possible ways to study the topic “Lagrange equation”. According to this method, the content of this topic should be disclosed through the following questions: generalized coordinates, generalized forces; generalized speeds and their units of measurement. We can state that the given method allows students to form a sufficiently deep and stable understanding of the concepts of “generalized coordinates”, “generalized forces”, “generalized velocities” and gives an idea of using the Lagrange equation for solving tasks that are difficult to solve using only Newton's laws. Further research will be related to the study of the methodological features of the study of the Lagrange equation on the basis of the integral variational principle of Hamilton-Ostrogradsky. Key words: future physics teachers, classical mechanics, generalized coordinates, generalized forces; generalized speeds, Lagrange equation.

Foundations of higher-order variational theory on Grassmann fibrations

A setting for higher-order global variational analysis on Grassmann fibrations is presented. The integral variational principles for one-dimensional immersed submanifolds are introduced by means of differential 1-forms with specific properties, similar to the Lepage forms from the variational calculus on fibred manifolds. Prolongations of immersions and vector fields to the Grassmann fibrations are defined as a geometric tool for the variations of immersions, and the first variation formula in the infinitesimal form is derived. Its consequences, the Euler–Lagrange equations for submanifolds and the Noether theorem on invariant variational functionals are proved. Examples clarifying the meaning of the Noether theorem in the context of variational principles for submanifolds are given.

The integral variational principles for embedded variation identity of high-order nonholonomic constrained systems

In this article, from the integral variational principles for embedded variation identity of high-order nonholonomic constrained systems, three kinds of dynamics for high-order nonholonomic constrained systems are obtained, including the vakonomic dynamical model, Lagrange-d'Alembert model and a new one if utilizing respectively three kinds of conditional variation to them. And the integral variational principles for embedded variation identity of high-order nonholonomic constrained systems is also fitted for the general nonholonomic systems when the constrained equation is reduced to a first-order one. Then, the vakonomic dynamic, Chetaev dynamics and a new model of general nonholonomic systems can also be obtained. Finally, two illustrated examples are used to verify the validity of the theory.

On variational formulations for functional differential equations

Necessary and sufficient conditions for the existence of integral variational principles for boundary value problems for given ordinary and partial functional differential equations are obtained. Examples are given illustrating the results.

UNIFIED EXPRESSIONS OF ALL INTEGRAL VARIATIONAL PRINCIPLES

In terms of the quantitative causal principle, this paper obtains a general variational principle, gives unified expressions of the general, Hamilton, Voss, Hölder, Maupertuis–Lagrange variational principles of integral style, the invariant quantities of the general, Voss, Hölder, Maupertuis–Lagrange variational principles are given, finally the Noether conservation charges of the general, Voss, Hölder, Maupertuis–Lagrange variational principles are deduced, and the intrinsic relations among the invariant quantities and the Noether conservation charges of all the integral variational principles are achieved.

Unification of different integral variational principles

In terms of a mathematical expression of the quantitative causal principle, this paper gives a unification of Hamilton, Voss, Hlder, Maupertuis-Lagrange varia tional principles of integral style of the second-order Lagrangians, and finds t he intrinsic relations among all the different integral variational principles. It is proved that f0= 0 is just the result satisfying the quantitati ve ca usal principle． The Noether conservation charges of Hamilton, Voss, Hlder, Ma upertuis-Lagrange variational principles are shown up, and the intrinsic relatio ns among the Noether conservation charges of all the integral variational princ iples are discovered. Our investigations make the expressions of the past scrappy numerous variational principles be unified into a relative consistent system o f all the variational principles in terms of the quantitative causal principle, and show that all the variational principles become deductions of the quantitat ive causal principle.

Variational Calculation of the Period of Nonlinear Oscillators

The problem of calculating the period of second order nonlinear autonomous oscillators is formulated as an eigenvalue problem. We show that the period can be obtained from two integral variational principles dual to each other. Upper and lower bounds on the period can be obtained to any desired degree of accuracy. The results are illustrated by an application to the Duffing equation.

Oscillatory dynamics of inviscid planar liquid sheets

One-dimensional models of inviscid, planar liquid sheets surrounded by dynamically passive gases are obtained by means of perturbation methods, integral formulations, series expansions and variational principles. It is shown that integral formulations, series expansions and variational principles provide analogous one-dimensional models under certain approximations for the velocity and pressure fields. The equations for slender and thin sheets obtained from integral formulations are shown to be asymptotically equivalent to the exact equations of inviscid, planar liquid membranes. Algebraic and differential methods are employed to determine the singularities of the steady equations obtained from integral formulations, and indicate that the liquid does not leave the nozzle with an angle equal to that of the exit if the Weber number is equal to or less than one. Numerical studies of the time-dependent governing equations are presented in order to illustrate the nonlinear dynamics and bifurcations of confined, inviscid, planar liquid sheets when they are subjected to time-dependent pressure differences or gases are injected on either side of the sheet, in the absence of heat and mass transfer between the gases and the liquid.

Integral variational principles in Poincaré and Chetayev variables

A holonomic mechanical system with k degrees of freedom is considered, its state being characterized by n ⩾ k defining coordinates, p < k Poincaré parameters [1] and k - p Chetayev parameters [2]. In these variables, generalized Routh equations are introduced and expressions are given for the integral variational principles of Hamilton-Ostrogradskii and Hamilton (the third form), as well as Hölder's principle and the Lagrange and Jacobi versions of the principle of least action.

Influence of dimensionality on solutions of diffraction problems in open waveguides

We compare the reflection characteristics of surface (guided) modes scattering from the abrupt termination of a planar dielectric waveguide or an optical fiber with an elongated cross section. Analytical expressions are derived for the reflection coefficients of slightly slowed-down fundametnal modes. The analysis is performed in the scalar approximation with the use of the integral equation method and variational principles. The difference between the characteristics of the problems considered is shown to be closely related to the analytic properties of eigenfunction sets of the waveguides.

Variational principles of second, first and intermediate kinds for non-holonomic mechanics

The differential variational principles of second kind for non-holonomic mechanics are given, from which a number of integral variational principles of second kind are set up. From the latter, the general relation of δq′-δq and the general form of integral variational principles of the first kind and intermediate kinds are derived. Thus not only all previous relations of δq′-δq and integral variational principles are unified but also the existance of the variational principles of intermediate kinds are pointed out.

On some important problems in analytical dynamics of non-holonomic systems

By using deductive method, Chetaev condition is derived in this paper. We point out that the processes of variation and differentiation are not permutable in non-holonomic dynamics is a misunderstanding. The paper gives two, classical relations of non-holonomic systems and points out integral variational principles can be applied in non-holonomic systems.

Qualitative Convergence in the Discrete Approximation of the Euler Problem∗

ABSTRACT There are two mathematically rigorous ways to derive Euler's differential equation of the elastica. The first is to start from integral rules and use variational principles, whereas the second is to regard the continuous rod as a limit of a discrete sequence of elastically connected rigid elements when the length of the elements decreases to zero. Discrete models of the Euler buckling problem are investigated. The global number s of solutions of the boundary-value problem is expressed as a function of the number of elements in the discrete model, s = s(n), at constant loading P. The functions s (n) are described by the characteristic parameters n 1 and n 2, and graphs of n 1(P) and n 2(P) are plotted. Observations related to these diagrams reveal interesting features in the behavior of the discrete model, from the point of view of both theory and application.

Variational principles for a nonreacting, nonviscous mixture

Integral variational principles for the statics of a nonreacting, nonviscous fluid mixture are presented, both for open and closed systems. It is demonstrated that the use of Gibbs’ free energy as ‘‘density function’’ gives the appropriate field equations only if a peculiar choice for constitutive equations is made: mass and heat fluxes must be linear in gradients of chemical potential and temperature.

Generalized integral variational principles of phenomenological thermodynamics of irreversible processes, and the nature of variation of thermodynamic action

Generalized integral variational principles of phenomenological thermodynamics of irreversible processes (PTIP), as formulated by the present author, are stated.

On the Lagrange and Jacobi principles for nonholonomic systems: PMM vol.43, no.4, 1979, pp. 583–590

It was shown in [1–4] that the integral variational principles of Hamilton, Lagrange, and Jacobi are valid for holonomic, as well as nonholonomic systems, although in the case of the latter the comparison curves do not, generally, satisfy the nonintegrable constraint equations. Hence, generally speaking, they are not principles of stationary action for nonholonomic systems. The necessary and sufficient conditions of existence of solutions of the equations of motion of nonholonomic systems among solutions of Euler's equations of the Lagrange problem were established in [5] for the Hamilton principle, i.e. when in the first approximation it is the principle of stationary action. The similar problem for the Lagrange and the Jacobi principles is solved here for mechanical systems subjected to nonholonomic stationary constraints homogeneous with respect to generalized velocities, and acted upon by potential forces defined by derivatives of the generalized force function. Necessary and sufficient conditions for the Lagrange and Jacobi principles to be principles of stationary action are established. These conditions coincide with those in [5]. Conditions under which the theorem and the energy integral of systems subjected to ideal nonlinear nonholonomic constraints are also formulated, and conditions under which real displacements of a nonholonomic system can be found among possible displacements are indicated.

Variational principles for nonstationary heat conduction problems

Integral variational principles are proposed for nonstationary heat conduction problems. These lead to integration of the wave equation of heat conduction or the Fourier heat conduction equation with the initial and boundary conditions.