Lagrange Multiplier Formalism Research Articles

Field redefinition invariant Lagrange multiplier formalism

In this paper, we propose a field redefinition invariant Lagrange multiplier (LM) formalism in which new ghost-like fields, analogous to Lee–Yang ghosts, are introduced. These ghost fields are required to restore the field redefinition invariance of the standard path integral of the LM theory and, at the same time, to cancel the additional contributions due to the LM fields. We argue that the extra degrees of freedom in the standard LM formalism, coming from the LM fields, should cancel against the degrees of freedom of the ghost fields. Hence, in the field redefinition invariant formalism the doubling of degrees of freedom, associated with the LM fields, is absent.

Free vibration of a sagged cable with attached discrete elements

An algorithm is presented to analyze the free vibration in a system composed of a cable with discrete elements, e.g., a concentrated mass, a translational spring, and a harmonic oscillator. The vibrations in the cable are modeled and analyzed with the Lagrange multiplier formalism. Some fragments of the investigated structure are modeled with continuously distributed parameters, while the other fragments of the structure are modeled with discrete elements. In this case, the linear model of a cable with a small sag serves as a continuous model, while the elements, e.g., a translational spring, a concentrated mass, and a harmonic oscillator, serve as the discrete elements. The method is based on the analytical solutions in relation to the constituent elements, which, when once derived, can be used to formulate the equations describing various complex systems compatible with an actual structure. The numerical analysis shows that, the method proposed in this paper can be successfully used to select the optimal parameters of a system composed of a cable with discrete elements, e.g., to detune the frequency resonance of some structures.

Phase-field theory of multicomponent incompressible Cahn-Hilliard liquids.

In this paper, a generalization of the Cahn-Hilliard theory of binary liquids is presented for multicomponent incompressible liquid mixtures. First, a thermodynamically consistent convection-diffusion-type dynamics is derived on the basis of the Lagrange multiplier formalism. Next, a generalization of the binary Cahn-Hilliard free-energy functional is presented for an arbitrary number of components, offering the utilization of independent pairwise equilibrium interfacial properties. We show that the equilibrium two-component interfaces minimize the functional, and we demonstrate that the energy penalization for multicomponent states increases strictly monotonously as a function of the number of components being present. We validate the model via equilibrium contact angle calculations in ternary and quaternary (four-component) systems. Simulations addressing liquid-flow-assisted spinodal decomposition in these systems are also presented.

VIBRATION MODEL AND ANALYSIS OF THREE-MEMBER TELESCOPIC BOOM WITH HYDRAULIC CYLINDER FOR ITS RADIUS CHANGE

In this paper, the free vibration problem of a system consisting of a three-member type telescopic boom of the laboratory crane, the hydraulic cylinder of crane radius change and the elastic support system has been presented. The model has been formulated in order to represent the vibrations in the lifting plane of the laboratory crane. The transverse vibrations of the boom, longitudinal and transverse vibrations of the hydraulic cylinder have been considered in the coupled model of the system taking into account the elasticity of the support system. In the computational model, the analyzed system has been replaced by the discrete-continuous model. The free vibration problem of the analyzed system is formulated and solved on the basis of Lagrange multiplier formalism. The experimental modal analysis is done and the genetic algorithm worked out with the aim of identifying spring constants of the support system elements. The analysis of the influence of the hydraulic cylinder and the length of the telescopic boom on the free vibration frequencies is carried out on the basis of the identified vibration model of the considered system. The examples of the experimental and numerical results are presented.

Discrete model of vibration of truck crane telescopic boom with consideration of the hydraulic cylinder of crane radius change in the rotary plane

In this paper the one degree of freedom discrete model representing the vibration of the telescopic boom of truck crane in the rotary plane has been presented. In the model the influence of the hydraulic cylinder of crane radius change has been considered. The formulation and solution of the problem has been based on the Lagrange multiplier formalism. The mathematical relations describing the equivalent mass and spring stiffness of the discrete model have been derived. The mathematical formulation is completed by the exemplary numerical results.

Nonextensive normalized treatment of Clausius equation

We revisit the first law of thermodynamics for nonextensive statistics by recourse to the optimal Lagrange multipliers (OLM) formalism. We show firstly that information theory allows one to obtain thermodynamics' first law in a general ensemble by dealing just with variations in the density operator, without disturbing the system's Hamiltonian. After re-deriving Clausius equation we discuss it in some detail and show that for the Tsallis–Mendes–Plastino (TMP) normalization: (i) Nonextensive reversible processes connecting off-equilibrium states exist, (ii) for any extensive reversible process between equilibrium states one can find some other, equivalent nonextensive irreversible one, and (iii) in some circumstances, an extensive irreversible process can be assimilated to a nonextensive reversible one.

Vínculos não-holônomos e equações de Voronec

Será permitido, no contexto do método dos multiplicadores de Lagrange, supor que vínculos nao-holônomos já estão em vigor durante a construção da lagrangiana? Este procedimento, embora aplicado com sucesso ao problema da moeda rolante num livro recente, não tem validade geral, como demonstramos por meio de um contra-exemplo. Em muitos casos, o uso de vínculos nao-holônomos no processo de construção da lagrangiana é permitido, mas as equações de movimento corretas são as equações de Voronec.

Hyperfine coupling tensors for multi-configurational quasi-degenerate perturbation theory

Analytic calculation of the hyperfine coupling tensors for multi-configurational quasi-degenerate perturbation theory is developed based on the Lagrange multiplier formalism. Calculation of Lagrange multipliers is not required if the corresponding constraining conditions do not depend explicitly on the magnetic moments of the nuclei. Except for the explicit form of the one-electron perturbation operator, the derivation presented in this work is also applicable to other molecular properties for which the basis functions do not depend on the differentiation variable, and for which the dependence of the Hamiltonian is through a one-electron operator only.

Thermodynamics’ zeroth law in a nonextensive scenario

We show how to reconcile Tsallis’ thermostatistics with thermodynamics’ zeroth law, by recourse to the so-called optimal Lagrange multipliers formalism. The central concept is that of not identifying in the usual fashion the inverse temperature with the Lagrange multiplier associated to the internal energy. Our analysis provides one with compatibility conditions between the additivity of the internal energy and the pseudo-additivity of the generalized entropy. With regards to the first law of thermodynamics, a generalization of Clausius’ equation is advanced.

Classical gas in nonextensive optimal Lagrange multipliers formalism

Based on the prescription termed the optimal Lagrange multipliers formalism for extremizing the Tsallis entropy indexed by q, it is shown that some key aspects of the treatment of the classical gas problem such as the internal energy and energy correlation are formally identical in both the nonextensive q≠1 and extensive q=1 cases.

Equipartition and virial theorems in a nonextensive optimal Lagrange multipliers scenario

We revisit some topics of classical thermostatistics from the perspective of the nonextensive optimal Lagrange multipliers formalism (OLM), a recently introduced technique for dealing with the maximization of Tsallis' information measure. It is shown that equipartition and virial theorems can be reproduced by Tsallis' nonextensive formalism independently of the value of the nonextensivity index.

A NOVEL APPROACH TO DETERMINE THE FREQUENCY EQUATIONS OF COMBINED DYNAMICAL SYSTEMS

A novel approach is presented which can be used to reduce the generalized eigenvalue problem associated with the free vibration of a linear elastic structure carrying lumped elements atsdistinct locations. UsingNcomponent modes in the assumed-modes method, the free vibration of such a combined dynamical system is governed by the solution of a generalized eigenvalue problem of orderN×N, whose stiffness and mass matrices consist of diagonal matrices modified by a total ofsrank one matrices, wherescorresponds to the number of attachment points. This generalized eigenvalue problem can be manipulated such that the natural frequencies governing free vibration can be calculated instead by solving a much smaller characteristic determinant of orders×s. Interestingly enough, this smaller and simpler characteristic determinant can also be obtained by using the Lagrange multipliers formalism in conjunction with Lagrange's equations.

FREE VIBRATIONS OF UNIFORM TIMOSHENKO BEAMS WITH LUMPED ATTACHMENTS

The concise eigenvalue equation for the free vibration of a combined dynamical system, obtained by means of the Lagrange multipliers formalism, can also be extracted by using the more straightforward and simpler assumed modes method

FREE VIBRATIONS OF UNIFORM TIMOSHENKO BEAMS WITH ATTACHMENTS

The problem of free transverse vibrations of Timoshenko beams with attachments like translational and rotational springs, concentrated mass including the moment of inertia, linear undamped oscillators and additional supports is considered. The frequency equation for the combined system is derived by means of the Lagrange multiplier formalism. The exact solution of the free vibration problem of the beam without attachments is taken into account for the formulation of the free vibration problem of the combined system. Numerical examples show the separate or coupling influences of the additional elements on the combined system's frequencies. The comparison of results obtained by using the present approach with results of the exact solution indicates a good agreement.

On hyperelasticity with internal constraints

By requiring the constitutive equation for the specific internal energy to be such that energy is balanced for all motions compatible with the internal constraint, we are able to infer the exitence and the direction of the reactive stress as well as the usual stress relation for the active stress. In contrast with previous work along this line, our analysis avoids the Lagrange multiplier formalism, and we need not assume that the internal energy response function is extendable off the constraint manifold.

Quantisation in a nonlinear gauge and Feynman rules

Certain aspects of the use of a non-linear gauge fixing condition in SU(N) gauge field theory are studied. The Faddeev-Popov ghost Lagrangian is obtained systematically using two different but related methods. Feynman rules are explicitly derived from the generating functional for the full Lagrangian in non-linear gauge. The tree-level amplitude for gauge field-gauge field scattering is shown to be independent of the parameter signalling the non-linearity of the gauge fixing condition and hence the same as the one in the linear covariant (Lorentz) gauge. The full Lagrangian is shown to have BRS invariance and the explicit form for BRS transformations is given. Using this, the corresponding Slavnov-Taylor identities are derived. The Lagrange multiplier formalism is used to derive the ghost Lagrangian independently and shown to be the same as that obtained by the gauge variation of the gauge fixing condition. The corresponding situation in Abelian gauge theories is reviewed. The coupling of the ghosts with gauge fields and the Gribov ambiguity are discussed.

The failure of the usual gauge formalism in a class of gauge conditions

For a class of gauge conditions the usual formalism for non-Abelian gauge theories leads to violation of unitarity. In contrast, the Lagrange multiplier formalism allows us to construct an adequate fictitious Lagrangian. The theory so obtained satisfies unitarity.

Explicit calculationvs. formal proof concerning unitarity violation of unified theories in Wu’s gauge

A formal proof and explicit check of unitarity for non-Abelian gauge theories with a particular bilinear gauge condition are presented. The theories are constructed within the Lagrange-multiplier formalism. It is concluded that the usual formalism for non-Abelian gauge theories is not completely satisfied because it leads to violation of unitarity when one uses the Wu-gauge condition. The Lagrange-multiplier formalism can reveal the stability of gauge conditions, if any, and has more general validity than the usual formalism. (JFP)

Density of states of tight-binding disordered systems

Rigorous upper and lower bounds to the integrated density of states of systems described within the tight-binding scheme are calculated using a Lagrange multipliers formalism which is valid for both equality and inequality constraints. Upper and lower bounds to the density of states are calculated as well; in this case, the constraints are the first few moments of the density of states and a bounded-below derivative of the density of states function. The bounds have been calculated for an oversimplified model of a binary alloy and for disordered magnetic insulators described within the Hubbard-model framework. Since the bounds are rather narrow, a great deal of information can be obtained from them. Upper and lower bounds are obtained to the density of states at the band tail energies in the antiferromagnetic arrangement of spins case. In the other cases studied, only upper bounds are obtained at these energies. Rigorous averages of the density of states and of the integrated density of states have also been obtained in terms of the moments of the density of states.

DYNAMIC PROGRAMMING AND LAGRANGE MULTIPLIERS

Abstract : In this paper it is shown that a combination of the classical Lagrange multiplier formalism and the functional equation technique of dynamic programming enables a number of types of variational problems involving the computation and tabulation of functions of M variables to be treated by computing first sequences of functions of K variables, and then sequences of functions of M--K variables, where K may be chosen within the range 1 or = K or = M-1. The choice of K depends upon the process This reduction in the dimensionality of the functions involved is equivalent to an increase in the capability of modern digital computers as far as dynamic programming processes are concerned.