Method Of Lagrangians Research Articles

Parallel iterative methods for variational integration applied to navigation problems

Abstract Discrete variational methods have shown an excellent performance in numerical simulations of different mechanical systems. In this paper, we introduce an iterative method for discrete variational methods appropriate for boundary value problems. More concretely, we explore a parallelization strategy that leverages the power of multicore CPUs and GPUs (graphics cards). We study this parallel method for first-order and second-order Lagrangians and we illustrate its excellent behavior in some interesting applications, namely Zermelo’s navigation problem, a fuel-optimal navigation problem, and an interpolation problem.

Thermodynamics of ferromagnetic spin chains in a magnetic field: Impact of the spin–wave interaction

The thermodynamic properties of ferromagnetic spin chains have been the subject of many publications. Still, the problem of how the spin–wave interaction manifest itself in these low-temperature series has been neglected. Using the method of effective Lagrangians, we explicitly evaluate the partition function of ferromagnetic spin chains at low temperatures and in the presence of a magnetic field up to three loops in the perturbative expansion where the spin–wave interaction sets in. We discuss in detail the renormalization and the numerical evaluation of a particular three-loop graph and derive the low-temperature series for the free energy density, energy density, heat capacity, entropy density, as well as the magnetization and the susceptibility. In the low-temperature expansion for the free energy density, the spin–wave interaction starts manifesting itself at order T5/2. In the pressure, the coefficient of the T5/2-term is positive, indicating that the spin–wave interaction is repulsive. While it is straightforward to go up to three-loop order in the effective loop expansion, the analogous calculation on the basis of conventional condensed matter methods, such as spin–wave theory, appears to be beyond reach.

Low-temperature properties of two-dimensional ideal ferromagnets

The manifestation of the spin-wave interaction in the low-temperature series of the partition function has been investigated extensively over more than seven decades in the case of the three-dimensional ferromagnet. Surprisingly, the same problem regarding ferromagnets in two spatial dimensions, to the best of our knowledge, has never been addressed in a systematic way so far. In the present paper the low-temperature properties of two-dimensional ideal ferromagnets are analyzed within the model-independent method of effective Lagrangians. The low-temperature expansion of the partition function is evaluated up to two-loop order and the general structure of this series is discussed, including the effect of a weak external magnetic field. Our results apply to two-dimensional ideal ferromagnets which exhibit a spontaneously broken spin rotation symmetry O(3) $\to $ O(2) and are defined on a square, honeycomb, triangular or Kagom\'e lattice. Remarkably, the spin-wave interaction only sets in at three-loop order. In particular, there is no interaction term of order $T^3$ in the low-temperature series for the free energy density. This is the analog of the statement that, in the case of three-dimensional ferromagnets, there is no interaction term of order $T^4$ in the free energy density. We also provide a careful discussion of the implications of the Mermin-Wagner theorem in the present context and thereby put our low-temperature expansions on safe grounds.

Energy Content of Colliding Plane Waves Using Approximate Noether Symmetries

We study the energy content of colliding plane waves using approximate Noether symmetries. For this purpose, we use the approximate Lie symmetry method for Lagrangians for differential equations. We formulate the first-order perturbed Lagrangian for colliding plane electromagnetic and gravitational waves. In both cases, we show that no nontrivial first-order approximate symmetry generator exists.

Formalismo de Hamilton-Jacobi à la Carathéodory. Parte 2: sistemas singulares

Na segunda parte do artigo publicado na Revista Brasileira de Ensino de Física 29, 393 (2007) trataremos do formalismo de Hamilton-Jacobi para sistemas singulares. Mostraremos como é possível generalizar o procedimento de Carathéodory para Lagrangianas cuja matriz Hessiana é singular e construir um procedimento de análise de vínculos a partir desse formalismo.

General formulas for invariant functions describing generalized γN → γN reactions within the method of effective Lagrangians

The crossed channels of generalized γN → γN reactions are considered. The coefficients in the transformation from independent helicity amplitudes to invariant functions are calculated. Explicit expressions for the invariant functions are derived with allowance for the contribution from the Born diagrams in the s, u, and t channels and the diagrams for six resonances in the s and u channels. It is shown that the calculated invariant functions satisfy the crossing-symmetry requirements.

A dynamic Lagrangian frequency–time method for the vibration of dry-friction-damped systems

A new frequency–time domain procedure, the dynamic Lagrangian mixed frequency–time method (DLFT), is proposed to calculate the non-linear steady state response to periodic excitation of structural systems subject to dry friction damping. In this formulation, the dynamic Lagrangians are defined as the non-linear contact forces obtained from the equations of motion in the frequency domain, with the adjunction of a penalization on the difference between the interface displacements calculate by the non-linear solver in the frequency domain and those calculated in the time domain from the non-linear contact forces, thus accounting for Coulomb friction and non-penetration conditions. The dynamic Lagrangians allow one to solve for the non-linear forces between two points in contact without using artifacts such as springs. The new DLFT method is thus particularly well suited to handling finite element models of structures in frictional contact, as it does not require a special model for the contact interface. Dynamic Lagrangians are also better suited to frequency-domain friction problems than the traditional time-domain method of augmented Lagrangians. Furthermore, a reduction of the non-linear system to relative interface displacements is introduced to decrease the computation time. The DLFT method is validated for a beam in contact with a flexible dry friction element connected to ground, for frictional constraints that feature two-dimensional relative motion. Results are also obtained for a large-scale structural system with a large number of one-dimensional dry-friction dampers. The DLFT method is shown to be accurate and fast, and it does not suffer from convergence problems, at least in the examples studied.

Averaged Lagrangians and the mean effects of fluctuations in ideal fluid dynamics

We begin by placing the generalized Lagrangian mean (GLM) equations for a compressible adiabatic fluid into the Euler–Poincaré (EP) variational framework of fluid dynamics, for an averaged Lagrangian. We then state the EP Averaging Result—that GLM equations arise from GLM Hamilton’s principles in the EP framework. Next, we derive a new set of approximate small-amplitude GLM equations ( gℓ m equations) at second order in the fluctuating displacement of a Lagrangian trajectory from its mean position. These equations express the linear and nonlinear back-reaction effects on the Eulerian mean fluid quantities by the fluctuating displacements of the Lagrangian trajectories in terms of their Eulerian second moments. The derivation of the gℓ m equations uses the linearized relations between Eulerian and Lagrangian fluctuations, in the tradition of Lagrangian stability analysis for fluids. The gℓ m derivation also uses the method of averaged Lagrangians, in the tradition of wave, mean flow interaction (WMFI). The gℓ m EP motion equations for compressible and incompressible ideal fluids are compared with the Euler-alpha turbulence closure equations. An alpha model is a GLM (or gℓ m) fluid theory with a Taylor hypothesis closure (THC). Such closures are based on the linearized fluctuation relations that determine the dynamics of the Lagrangian statistical quantities in the Euler-alpha closure equations. We use the EP Averaging Result to bridge between the GLM equations and the Euler-alpha closure equations. Hence, combining the small-amplitude approximation with THC yields in new gℓm turbulence closure equations for compressible fluids in the EP variational framework.

Lagrangian averages, averaged Lagrangians, and the mean effects of fluctuations in fluid dynamics.

We begin by placing the generalized Lagrangian mean (GLM) equations for a compressible adiabatic fluid into the Euler-Poincare (EP) variational framework of fluid dynamics, for an averaged Lagrangian. This is the Lagrangian averaged Euler-Poincare (LAEP) theorem. Next, we derive a set of approximate small amplitude GLM equations (glm equations) at second order in the fluctuating displacement of a Lagrangian trajectory from its mean position. These equations express the linear and nonlinear back-reaction effects on the Eulerian mean fluid quantities by the fluctuating displacements of the Lagrangian trajectories in terms of their Eulerian second moments. The derivation of the glm equations uses the linearized relations between Eulerian and Lagrangian fluctuations, in the tradition of Lagrangian stability analysis for fluids. The glm derivation also uses the method of averaged Lagrangians, in the tradition of wave, mean flow interaction. Next, the new glm EP motion equations for incompressible ideal fluids are compared with the Euler-alpha turbulence closure equations. An alpha model is a GLM (or glm) fluid theory with a Taylor hypothesis closure. Such closures are based on the linearized fluctuation relations that determine the dynamics of the Lagrangian statistical quantities in the Euler-alpha equations. Thus, by using the LAEP theorem, we bridge between the GLM equations and the Euler-alpha closure equations, through the small-amplitude glm approximation in the EP variational framework. We conclude by highlighting a new application of the GLM, glm, and alpha-model results for Lagrangian averaged ideal magnetohydrodynamics. (c) 2002 American Institute of Physics.

Decay τ − → φπ − ν τ and ω−φ mixing

The decay process τ− → φπ−ντ is investigated on the basis of the method of chiral phenomenological Lagrangians. It is shown that the calculated value of the decay probability is very sensitive to variations in the angle of ω−φ mixing. The resulting value of this probability is compared with available experimental data and with the results of other theoretical calculations.

QCD-like theories at finite baryon and isospin density

We use two-color QCD as a model to study the effects of the simultaneous presence of chemical potentials for isospin charge ${\ensuremath{\mu}}_{I}$ and for baryon number ${\ensuremath{\mu}}_{B}.$ We determine the phase diagrams for two and four flavor theories using the method of effective chiral Lagrangians at low densities and weak-coupling perturbation theory at high densities. We determine the values of various condensates and densities as well as the spectrum of excitations as functions of ${\ensuremath{\mu}}_{I}$ and ${\ensuremath{\mu}}_{B}.$ A similar analysis of QCD with quarks in the adjoint representation is also presented. Our results can be of relevance for lattice simulations of these theories. We predict a phase of inhomogeneous condensation (Fulde-Ferrel-Larkin-Ovchinnikov phase) in the two color, two flavor theory, while we do not expect it in the four flavor case or in other realizations of QCD with a positive measure.

Auto-Parallel Equation as Euler-Lagrange's Equation over Spaces with Affine Connections and Metrics

The auto-parallel equation over spaces with affine connections and metrics is considered as a result of the application of the method of Lagrangians with covariant derivatives (MLCD) on a given Lagrangian density.

Numerical simulation of electrorheological fluids based on an extended Bingham model

In the framework of the macroscopic simulation of electrorheological fluids, we present an extension of the classical Bingham model which goes beyond pure shear flows and thus enables the simulation of settings in more complex geometries. Emphasis is on the numerical solution of the resulting nonsmooth minimization problem. We propose the method of augmented Lagrangians combined with an operator-splitting technique which allows to confine the nonlinearity to local, low-dimensional problems. Numerical results are given that illustrate the electrorheological effects for various shear rates and electric field strengths in case of an electrorheological suspension rotating between two revolving cylinders.

Lagrangian Theory of Tensor Fields over Spaces with Contravariant and Covariant Affine Connections and Metrics and Its Application to Einstein's Theory of Gravitation in V 4 Spaces

The Lagrangian formalism for tensor fields over differentiable manifolds with contravariant and covariant affine connections (whose components differ not only by sign) and a metric \([(\bar L_n ,g) - spaces]\) is considered. The functional, the Lie, the covariant, and the total variations of a Lagrangian density, depending on components of tensor fields (with finite rank) and their first and second covariant derivatives, are established. A variation operator is determined and the corollaries of its commutation relations with the covariant and the Lie differential operators are found. The canonical (common) method of Lagrangians with partial derivatives (MLPD) and the method of Lagrangians with covariant derivatives (MLCD) are outlined. They differ each other by the commutation relations the variation operator has to obey with the covariant and the Lie differential operator. The covariant Euler–Lagrange equations are found on the basis of the MLCD. The energy-momentum tensors are considered on the basis of the Lie variation and the covariant Noether identities.

On theτ→VPντdecays

The $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\tau}}\mathrm{VP}{\ensuremath{\nu}}_{\ensuremath{\tau}}$ decays of the $\ensuremath{\tau}$ lepton are studied using the method of phenomenological chiral Lagrangians. The expressions of weak hadronic currents between pseudoscalar and vector meson states and the strong interaction Lagrangian between axial-vector, vector, and pseudoscalar mesons are obtained. Calculated partial widths for these decays are compared with the available theoretical and experimental data.

EINSTEIN'S THEORY OF GRAVITATION AS A LAGRANGIAN THEORY FOR TENSOR FIELDS

Einstein's theory of gravitation (ETG) is considered as a Lagrangian theory of tensor fields over (pseudo) Riemannian spaces without torsion (Vn spaces, n=4) by means of the method of Lagrangians with covariant derivarives (MLCD). In a trivial manner Euler–Lagrange's equations as Einstein's equations are obtained. The corresponding energy–momentum tensors (EMT's) are found for the standard for the ETG Lagrangian invariant on the basis of the covariant Noether identities. The symmetric energy–momentum tensor of Hilbert appears as an element irrelevant to the whole scheme of the considered Lagrangian thoery of tensor fields over Vn spaces despite of the fact that it has some elements of the structure of the variational EMT of Euler–Lagrange. The notion of the active gravitational rest mast density is related to the variational EMT of Euler–Lagrange and on this basis to a certain extent to the EMT of Hilbert.

Modified barrier functions (theory and methods)

The nonlinear rescaling principle employs monotone and sufficiently smooth functions to transform the constraints and/or the objective function into an equivalent problem, the classical Lagrangian which has important properties on the primal and the dual spaces. The application of the nonlinear rescaling principle to constrained optimization problems leads to a class of modified barrier functions (MBF's) and MBF Methods (MBFM's). Being classical Lagrangians (CL's) for an equivalent problem, the MBF's combine the best properties of the CL's and classical barrier functions (CBF's) but at the same time are free of their most essential deficiencies. Due to the excellent MBF properties, new characteristics of the dual pair convex programming problems have been found and the duality theory for nonconvex constrained optimization has been developed. The MBFM have up to a superlinear rate of convergence and are to the classical barrier functions (CBF's) method as the Multipliers Method for Augmented Lagrangians is to the Classical Penalty Function Method. Based on the dual theory associated with MBF, the method for the simultaneous solution of the dual pair convex programming problems with up to quadratic rates of convergence have been developed. The application of the MBF to linear (LP) and quadratic (QP) programming leads to a new type of multipliers methods which have a much better rate of convergence under lower computational complexity at each step as compared to the CBF methods. The numerical realization of the MBFM leads to the Newton Modified Barrier Method (NMBM). The excellent MBF properties allow us to discover that for any nondegenerate constrained optimization problem, there exists a "hot" start, from which the NMBM has a better rate of convergence, a better complexity bound, and is more stable than the interior point methods, which are based on the classical barrier functions.

An augmented lagrangian treatment of contact problems involving friction

A framework is presented within which the method of augmented Lagrangians is readily applied to problems involving contact with friction. This method, which has enjoyed considerable success in the treatment of constrained minimization problems, has been previously applied to problems involving incompressible flow, incompressible elasticity of solids and even frictionless contact. An additional challenge to the method is provided by frictional contact problems governed by a Coulomb law, due to the special form taken by the frictional constraint. This paper describes a new extension of the augmented Lagrangian technique to frictional problems which is well-suited to finite element implementation. The proposed treatment inherits the traditional advantages of augmented Lagrangian techniques over penalty methods; namely, decreased ill-conditioning of governing equations, and essentially exact satisfaction of constraints with finite penalties. A set of numerical examples is presented in which the utility of the method is demonstrated even in the presence of finite deformations and inelasticity.

Invariant background field method for chiral lagrangians including Wess-Zumino terms

A chiral invariant background field method is introduced in order to study the structure of higher order terms in chiral perturbation theory. Explicit one-loop calculations are presented together with the construction of the most general O ( p 6) chiral lagrangian relevant for anomalous processes. The non-renormalization of the coefficient of the Wess-Zumino term in both two and four dimensions is discussed in general.

Invariant background field method for chiral lagrangians including Wess-Zumino terms: R. Akhoury and A. Alfakih. Randall Lab of Physics, University of Michigan, Ann Arbor, Michigan 48109-1120

Invariant background field method for chiral lagrangians including Wess-Zumino terms: R. Akhoury and A. Alfakih. Randall Lab of Physics, University of Michigan, Ann Arbor, Michigan 48109-1120