Euler-Lagrange Variational Principle Research Articles

Variational Estimation Methods for Sturm–Liouville Problems

In this paper, we are concerned with approach solutions for Sturm–Liouville problems (SLP) using variational problem (VP) formulation of regular SLP. The minimization problem (MP) is also set forth, and the connection between the solution of each formulation is then proved. Variational estimations (the variational equation associated through the Euler–Lagrange variational principle and Nehari’s method, shooting method and bisection method) and iterative variational methods (He’s method and HPM) for regular RSL are unitary presented in final part of the paper, which ends with applications.

Coupling effects of surface energy, strain gradient, and inertia gradient on the vibration behavior of small-scale beams

This paper develops three surface energy-enriched gradient elastic beam models, respectively, in the contexts of Euler-Bernoulli, Timoshenko, and Levinson-Brickford-Reddy kinematic hypotheses to investigate the coupling effects of surface energy, strain gradient, and inertia gradient on the vibration behavior of small-scale beams. Modified strain-inertia gradient elasticity theory and Gurtin-Murdoch surface elasticity theory are combined to capture three types of small-scale effects. The equations of motion and consistent boundary conditions are derived by using the Euler-Lagrange variational principle. To analyze the general free vibration problems of three non-classical beam models, we construct the related differential quadrature finite elements according to two kinds of differential quadrature-based geometric mapping schemes. The efficacy of our theoretical models and numerical solution methods is established by comparing the degenerated results with the reported ones. Finally, we apply the newly developed models to investigate the size-dependent behavior of freely vibrating small-scale beams. It is revealed that the coupling effects of three physical factors can result in not only the stiffening or softening characteristic of vibration frequencies but also the significant change in the vibration mode shapes. Besides, the introduction of the strain gradient effect may induce a boundary layer for small-scale beams with at least one end clamped.

Combined effects of surface energy and couple stress on the nonlinear bending of FG-CNTR nanobeams

This paper investigates the size-dependent nonlinear bending of functionally graded carbon nanotube-reinforced (FG-CNTR) nanobeams. Chen–Yao’s surface elasticity and modified couple stress theories are adopted to describe surface effects and couple stress effects, respectively. These nanobeams, in which the carbon nanotube (CNT)-reinforced phases are assumed to be distributed in a gradient along the thickness, are subjected to a uniform pressure and rest on a nonlinear elastic foundation. In accordance with the Euler–Lagrange variational principle, the governing equations and boundary conditions for the FG-CNTR nanobeams, which involve geometric nonlinearity due to the von Kármán strain relations, are obtained. Then, with the assistance of the two-step perturbation technique, the load-deflection relationship is determined for nanobeams subjected to simply supported (SS) and clamped–clamped (CC) boundary conditions. Finally, the impacts of various factors, including surface properties, characteristic material length, elastic foundation, geometric factors, layout type and volume fraction of CNTs, on the mechanical behaviors of CNT-based nanobeams are examined. The numerical results reveal that the combination of surface effects and couple stress helps to enhanceq the stiffness of the nanobeams. Furthermore, the size-dependent nonlinear bending of the FG-CNTR nanobeam is markedly affected by the content and layout type of the reinforcements.

Nonlinear Bending Analysis of Functionally Graded CNT-Reinforced Shallow Arches Placed on Elastic Foundations

This research explores the nonlinear bending behaviors of functionally graded carbon nanotube-reinforced (FG-CNTR) shallow arches with unmovable simply supported ends and clamped–clamped ends; these arches are subjected to a uniform radial pressure and rest on a nonlinear elastic foundation. The temperature-dependent material properties of the arches are considered. Within the framework of Reddy shear deformation theory possessing von Karman nonlinearity, the motion equations and boundary conditions for the FG-CNTR arches are determined by the Euler–Lagrange variational principle. Then, a two-step perturbation technique is adopted to determine the load–deflection relationship analytically. To verify the validity of the developed model and related perturbation solutions, a numerical investigation is conducted for shallow arches with five distribution patterns of carbon nanotube (CNT) reinforcements uniaxially aligned in the axial direction. Finally, the influences of various factors, including the elastic foundation, layout type, and volume fraction of CNTs and geometric factors, on the nonlinear behaviors of FG-CNTR shallow arches are examined. The obtained results show that the load–deflection curves exhibit less snap-through instability as the CNT volume fraction increases. The transverse shear stress versus the thickness of FG-CNTR shallow arches is markedly affected by the layout type and content of reinforcements.

Coupled forward-backward trajectory approach for nonequilibrium electron-ion dynamics

We introduce a simple ansatz for the wavefunction of a many-body system based on coupled forward and backward-propagating semiclassical trajectories. This method is primarily aimed at, but not limited to, treating nonequilibrium dynamics in electron-phonon systems. The time-evolution of the system is obtained from the Euler-Lagrange variational principle, and we show that this ansatz yields Ehrenfest mean field theory in the limit that the forward and backward trajectories are orthogonal, and in the limit that they coalesce. We investigate accuracy and performance of this method by simulating electronic relaxation in the spin-boson model and the Holstein model. Although this method involves only pairs of semiclassical trajectories, it shows a substantial improvement over mean field theory, capturing quantum coherence of nuclear dynamics as well as electron-nuclear correlations. This improvement is particularly evident in nonadiabatic systems, where the accuracy of this coupled trajectory method extends well beyond the perturbative electron-phonon coupling regime. This approach thus provides an attractive route forward to the ab-initio description of relaxation processes, such as thermalization, in condensed phase systems.

Solution of the five-layer beam bending problem using the calculus of variation

The paper deals with the problem of lateral bending for the five-layer panel with the lightweight filler that is reduced to the bending problem of a beam fixed at the end sections. To solve the differential equations describing the beam bending, the Euler-Lagrange variational principle is used.

Variational Approach for Resolving the Flow of Generalized Newtonian Fluids in Circular Pipes and Plane Slits

We use a generic and general variational method to obtain solutions to the flow of generalized Newtonian fluids through circular pipes and plane slits. The new method is not based on the use of the Euler-Lagrange variational principle and hence it is totally independent of our previous approach which is based on this principle. Instead, the method applies a very generic and general optimization approach which can be justified by the Dirichlet principle although this is not the only possible theoretical justification. The results that were obtained from the new method using nine types of fluid are in total agreement, within certain restrictions, with the results obtained from the traditional methods of fluid mechanics as well as the results obtained from the previous variational approach. In addition to being a useful method in its own for resolving the flow field in circular pipes and plane slits, the new variational method lends more support to the old variational method as well as for the use of variational principles in general to resolve the flow of generalized Newtonian fluids and obtain all the quantities of the flow field which include shear stress, local viscosity, rate of strain, speed profile, and volumetric flow rate.

Using the Euler–Lagrange variational principle to obtain flow relations for generalized Newtonian fluids

Euler-Lagrange variational principle is used to obtain analytical and numerical flow relations in cylindrical tubes. The method is based on minimizing the total stress in the flow duct using the fluid constitutive relation between stress and rate of strain. Newtonian and non-Newtonian fluid models; which include power law, Bingham, Herschel-Bulkley, Carreau and Cross; are used for demonstration.

Rigid supersymmetry with boundaries

We construct rigidly supersymmetric bulk-plus-boundary actions, both in $x$-space and in superspace. For each standard supersymmetric bulk action a minimal supersymmetric bulk-plus-boundary action follows from an extended $F$- or $D$-term formula. Additional separately supersymmetric boundary actions can be systematically constructed using co-dimension one multiplets (boundary superfields). We also discuss the orbit of boundary conditions which follow from the Euler-Lagrange variational principle.

Variational density matrices in quantum field theory at finite temperature and chemical potential.

I evaluate the Helmholtz free energy of finite temperature {lambda}{var_phi}{sup 4} theory, both real and complex, using a variational quadratic {ital ansatz} for the density matrix. Minimizing with respect to the variational parameters produces results identical to those obtained by summing the daisy and superdaisy diagrams. In the nonrelativistic limit this is equivalent to a Hartree-Fock mean field with an effective mass. Quartic terms are then included by means of a relativistic generalization of the hypernetted-chain approximation without exchange terms, called the {open_quote}{open_quote}direct approximation.{close_quote}{close_quote} In this way infinite groups of rings and ladders are summed, giving nonperturbative expressions for the internal energy and four-point function in terms of a small number of Dyson-like integral equations. An expression is obtained for the internal energy of a zero-temperature system in terms of only two variational parameters. Because the hypernetted-chain approximation preserves the Euler-Lagrange variational principle, minimizing the internal energy with respect to these parameters should provide a semiquantitative upper bound on the ground state energy of an interacting relativistic system at zero temperature. For the full finite temperature theory in the direct approximation, there are now three variational parameters and it is necessary to obtain the entropy in a approximation comparable tomore » that for the internal energy. This is done in an analogous manner to the separability approximation of nonrelativistic hypernetted-chain theory. Finally, an improvement on the direct approximation is attained by including exchange terms of all types. This proceeds along the lines of parquet summations, resulting in a set of integral equations that, when solved self-consistently, includes all series and parallel connections of direct and exchange diagrams. {copyright} {ital 1996 The American Physical Society.}« less

Green's formula and variational principles for cosmic-ray transport with application to rotating and shearing flows

Green's theorem and Green's formula for the diffusive cosmic-ray transport equation in relativistic flows are derived. Green's formula gives the solution of the transport equation in terms of the Green's function of the adjoint transport equation, and in terms of distributed sources throughout the region R of interest, plus terms involving the particle intensity and streaming on the boundary. The adjoint transport equation describes the time-reversed particle transport. An Euler-Lagrange variational principle is then obtained for both the mean scattering frame distribution function f, and its adjoint f(dagger). Variations of the variational functional with respect to f(dagger) yield the transport equation, whereas variations of f yield the adjoint transport equation. The variational principle, when combined with Noether's theorem, yields the conservation law associated with Green's theorem. An investigation of the transport equation for steady, azimuthal, rotating flows suggests the introduction of a new independent variable H to replace the comoving frame momentum variable p'. For the case of rigid rotating flows, H is conserved and is shown to be analogous to the Hamiltonian for a bead on a rigidly rotating wire. The variable H corresponds to a balance between the centrifugal force and the particle inertia in the rotating frame. The physical interpretation of H includes a discussion of nonrelativistic and special relativistic rotating flows as well as the cases of aziuthal, differentially rotating flows about Schwarzs-child and Kerr black holes. Green's formula is then applied to the problem of the acceleration of ultra-high-energy cosmic rays by galactic rotation. The model for galactic rotation assumes an angular velocity law Omega = Omega(sub 0)(omega(sub 0)/omega), where omega denotes radial distance from the axis of rotation. Green's functions for the galactic rotation problem are used to investigate the spectrum of accelerated particles arising from monoenergetic and truncated power-law sources. We conclude that it is possible to accelerate particles beyond the knee by galactic rotation, but not in sufficient number to adequately explain the observed spectrum.

Refined orthogonal variational treatment of continuum distorted waves

Two-state coupled equations within the continuum distorted-wave representation (CDW) are derived using the time-dependent second-order Euler-Lagrange variational principle of Sil (1980). Matrix elements are simplified using generalised non-orthogonal coordinates. It is shown that the original CDW theory of Cheshire (1964) is a variation-perturbation approximation, analogous to the Oppenheimer-Brinkman-Kramers approximation (OBK) in the travelling atomic eigenstate representation. It is also shown that unlike OBK the CDW wavefunctions are in general not normalised and the CDW wavefunction normalisation integrals are therefore calculated numerically using a complex Laplace transform. The implications of a variational CDW theory of ionisation are briefly considered.