Exact Lagrangian Research Articles

New method of deriving local energy- and momentum-conserving Maxwell-collisionless drift-kinetic and gyrokinetic theories: basic theory

In this paper we describe a relatively simple and transparent method of obtaining collisionless drift-kinetic, gyrokinetic and more general theories, including local charge, energy and momentum conservation laws. An important feature of the new formalism is, contrary to present-day theories, the exact gauge invariance, thus avoiding certain inconsistencies. The present paper starts with the introduction and proof of the correctness of a Lagrangian for combined Maxwell-kinetic theories in general coordinates as concerns the particle motion. The kinetic part of it is formulated in Eulerian form by means of the equations of motion in the form of Hamilton–Jacobi's equation, used only as a tool, and Dirac's constraint theory. Charge and current densities automatically distinguish between ‘particle-like’ (guiding-centre), polarization and magnetization contributions. This formalism is applied to averaging coordinates derived by a method similar to Kruskal's. Certain properties of the averaging coordinates, according to the basic requirements imposed on them, can be used to obtain a gyroangle independent Lagrangian, from which one can obtain a Lagrangian for the combined Maxwell-kinetic theories in a reduced phase space that is applicable to situations in which one is not interested in the dependence on some kind of gyroangle describing the gyromotion, whose treatment can, however, easily be added. The basic perturbation theory, which aims at obtaining averaging phase-space coordinates, is done solely within the Kruskal formalism in which only the electric and magnetic fields appear, but not the corresponding potentials. This formalism provides, in particular, information about the allowed amplitudes of fluctuations. The results are later used to obtain approximate expressions for the Lagrange functions in the drift-kinetic and the gyrokinetic ordering. For the definition of certain approximations to the exact Lagrangian the central principle is the exact gauge invariance. The terms of the zeroth and first orders needed for such an approximation are given. For the drift-kinetic ordering Littlejohn's Lagrangian is readily rederived and hence the drift-kinetic theory as obtained and investigated by the present authors in some previous work. Conservation laws and their mathematical structures corresponding to the gyroangle-independent Lagrangian are obtained in a following paper. There, in particular, a detailed description of how to obtain variations of gyroangle averaged quantities in the reduced phase space is given, and it is explained how these variations are used in a modified form of Noether's theory which observes exact gauge invariance.

Polarization of an electron-positron vacuum by a strong magnetic field with an allowance made for the anomalous magnetic moments of particles

Given the anomalous magnetic moments of electrons and positrons in the one-loop approximation, we calculate the exact Lagrangian of an intense constant magnetic field that replaces the Heisenberg-Euler Lagrangian in traditional quantum electrodynamics (QED). We have established that the derived generalization of the Lagrangian is real for arbitrary magnetic fields. In a weak field, the calculated Lagrangian matches the standard Heisenberg-Euler formula. In extremely strong fields, the field dependence of the Lagrangian completely disappears and the Lagrangian tends to a constant determined by the anomalous magnetic moments of the particles.

Effective lagrangian for 3d Script N = 4 SYM theories for any gauge group and monopole moduli spaces

We construct low energy effective Lagrangians for 3d N=4 supersymmetric Yang-Mills theory with any gauge group. They represent supersymmetric sigma models at hyper-Kahlerian manifolds of dimension 4r (r is the rang of the group). In the asymptotic region, perturbatively exact explicit expression for the metric are written. We establish the relationship of this metric with the TAUB-NUT metric describing the perturbatively exact effective Lagrangians for unitary groups and monopole moduli spaces: the former is obtained out of the latter by a proper hyper-Kahlerian reduction. We describe in details the reduction procedure for SO/Sp/G_2 gauge groups, where it can also be given a natural interpretation in D-brane language. We conjecture that the exact nonperturbative metrics can be obtained by a similar hyper-Kahlerian reduction from the corresponding multidimensional Atiyah-Hitchin metrics.

A model of adhesion coupled to contact and friction

An original law of adhesion is developed and coupled to the classical laws of unilateral contact and threshold friction. Its originality stems from the choice of an irreversible decohesion gap for internal variable, together with an adhesive tension for conjugate stress. Adhesion is modeled as a tension limit at zero gap and decohesion as a linear “plastic” softening over a finite gap, which makes the law more realistic than previous ones. The whole tribological law is formulated within the framework of standard generalized materials adapted to interfaces. For solution purposes, the law is next regularized using either the approximate penalty or the exact augmented Lagrangian method. The corresponding quasi-incremental minimisation and minimaximisation contact problems are formulated. The model is illustrated by the numerical simulations of the decohesion of a rigid parabolic punch on an elastic half-space and the pull-out of a fibre embeded in a matrix.

An Exact Augmented Lagrangian Function for Nonlinear Programming with Two-Sided Constraints

This paper is aimed toward the definition of a new exact augmented Lagrangian function for two-sided inequality constrained problems. The distinguishing feature of this augmented Lagrangian function is that it employs only one multiplier for each two-sided constraint. We prove that stationary points, local minimizers and global minimizers of the exact augmented Lagrangian function correspond exactly to KKT pairs, local solutions and global solutions of the constrained problem.

Classical Many-Body Problems Amenable to Exact Treatments. (Solvable and/or Integrable and/or Linearizable…) in One-, Two-, and Three-Dimensional Space

9R7. Classical Many-Body Problems Amenable to Exact Treatments. (Solvable and/or Integrable and/or Linearizable…) in One-, Two-, and Three-Dimensional Space. Lecture Notes in Physics, Vol m66. - F Calogero (Dept of Phys, Univ of Rome “La Sapienza,” p Aldo Moro, Rome, 00185, Italy). Springer-Verlag, Berlin. 2001. 749 pp. ISBN 3-540-41764-8. $79.95.Reviewed by M Pascal (Lab de Modelisation en Mec, Univ Pierre et Marie Curie, Tour 66, 4 Place Jussieu, Paris, 75252 Cedex 05, France).This book is concerned with integrable problems in classical mechanics (excluding quantam or relativistic mechanics). These problems are related to the motion of a system of particles acted upon by several kinds of forces in one-, two-, or three-dimensions. More general Hamiltonian systems (not associated with a Lagrangian function) are also considered. In most cases, integrable Hamiltonian systems mean Liouville integrable systems. In almost all examples dealing with the motion of a set of particles, the masses are the same for all particles. Several methods to find integrable problems are described, but the main part of the book is devoted to the Lax pair technique. This book provides a rather exhaustive survey of results obtained in the past by scientists such as Moser, Toda, Flashka, and also by the author himself.The book involves five chapters followed by several appendices and a rather wide list of references. Each chapter includes a great amount of examples and exercises. Due to the rather abstract topics studied in the book, no figures are included in it. The first chapter is a short introduction giving some basic knowledge about Newtonian equations of motion, Hamiltonian formalism, and Liouville integrable systems. The main part of the book is contained in Chapter 2, which deals with one-dimensional motions of particles along a line or a circle. Liouville integrable problems are found by using the Lax pair technique. Several examples are obtained by solving three kinds of functional equations. From these integrable cases, other integrable systems are deduced by means of several transformations like change of variables, duplication and so on. Another method introduces a time dependant polynomial of degree N with respect to the variable X; the N zeros of this polynomial are interpreted as the positions of N particles (with the same masses) on a line. Assuming that the polynomial is solution of a linear partial derivative equation, several solvable many body problems are found. An extension of this approach is also presented, using the exact Lagrangian interpolation method. In Chapter 3, a generalized formulation of this last method is formulated for spaces of arbitrary dimensions. Chapter 4 deals with integrable systems in the plane. The method is based on the idea that integrable problems in two-dimensional space can be deduced from integrable problems in one-dimension assuming that the equations of motion hold also for complex values of the variables: it is the so called technique of complexification. A survey of integrable systems in the plane obtained by this method is given at the end of the chapter. The last chapter presents integrable many body systems in three-dimensional space. The idea is to identify some solvable evolution equations governing a time dependant matrix with the motion equations of a set of particles in three-dimensions. Again, several examples are presented. As a conclusion, Classical Many-Body Problems Amenable to Exact Treatments…, is not intended for a very large audience, but it provides a rather exhaustive survey about integrable problems in some special cases of particle dynamics. It can be useful for researchers involved in celestial mechanics or can be used as a background for teaching an undergraduate course.

An Augmented Lagrangian Function with Improved Exactness Properties

In this paper we introduce a new exact augmented Lagrangian function for the solution of general nonlinear programming problems. For this Lagrangian function a complete equivalence between its unconstrained minimization on an open set and the solution of the original constrained problem can be established under mild assumptions and without requiring the boundedness of the feasible set of the constrained problem. Moreover we describe an unconstrained algorithmic model which is globally convergent toward KKT pairs of the original constrained problem. The algorithmic model can be endowed with a superlinear rate of convergence by a proper choice of the search direction in the unconstrained minimization, without requiring strict complementarity.

A Shifted-Barrier Primal-Dual Algorithm Model for Linearly ConstrainedOptimization Problems

In this paper we describe a Newton-type algorithm model for solving smooth constrained optimization problems with nonlinear objective function, general linear constraints and bounded variables. The algorithm model is based on the definition of a continuously differentiable exact merit function that follows an exact penalty approach for the box constraints and an exact augmented Lagrangian approach for the general linear constraints. Under very mild assumptions and without requiring the strict complementarity assumption, the algorithm model produces a sequence of pairs lx^k, λ^kr converging quadratically to a pair ({\bar{x},\bar\lambda}) where {\bar{x} satisfies the first order necessary conditions and {\bar\lambda} is a KKT multipliers vector associated to the linear constraints. As regards the behaviour of the sequence lx^kr alone, it is guaranteed that it converges at least superlinearly. At each iteration, the algorithm requires only the solution of a linear system that can be performed by means of conjugate gradient methods. Numerical experiments and comparison are reported.

Min-max dynamic response optimization of mechanical systems using approximate augmented Lagrangian

A globally convergent and efficient algorithm for min–max dynamic response optimization using the ALM method is presented. This algorithm employs an approximate augmented Lagrangian for line searches, and an exact augmented Lagrangian for finding search directions. This approximate augmented Lagrangian is composed of the linearized cost and constraint functions projected on the search direction. It is noted that an approximate penalty term has the same approximate Hessian as that of the Gauss–Newton method. This makes the approximate augmented Lagrangian to have almost second-order accuracy near the optimum. In the unconstrained optimization process of the proposed method, some modifications are also suggested to remove the possible distortion of optimization trajectory when the side constraints on design variables are separately handled. Also, the efficiencies of two treatments for handling a max-value cost function are discussed in the context of the proposed method. The numerical performance of the proposed method is investigated by solving three min–max dynamic response optimization problems and comparing the results with those in the literature. This comparison shows that the suggested algorithm is more efficient than those in the literature. Especially, it is found that the direct treatment for a max-value cost function is more efficient than the transformation treatment for all the design cases. © 1998 John Wiley & Sons, Ltd.

Lagrangians for plasmas in the drift-fluid approximation

For drift waves and related instabilities, conservation laws can play a crucial role. In an ideal theory these conservation laws are guaranteed when a Lagrangian can be found from which the equations for the various quantities result by Hamilton's principle. Such a Lagrangian for plasmas in the drift-fluid approximation was obtained by a heuristic method in a recent paper by Pfirsch and Correa-Restrepo. In the present paper the same Lagrangian is derived from the exact multifluid Lagrangian via an iterative approximation procedure which resembles the standard method usually applied to the equations of motion. That method, however, does not guarantee that all the conservation laws hold.

A Path Integral Approach to Option Pricing with Stochastic Volatility: Some Exact Results

The Black-Scholes formula for pricing options on stocks and other securities has been generalized by Merton and Garman to the case when stock volatility is stochastic. The derivation of the price of a security derivative with stochastic volatility is reviewed starting from the first principles of finance. The equation of Merton and Garman is then recast using the path integration technique of theoretical physics. The price of the stock option is shown to be the analogue of the Schrodinger wavefuction of quantum mechacnics and the exact Hamiltonian and Lagrangian of the system is obtained. The results of Hull and White are generalized results for pricing stock options for the general correlated case are derived.

Energy of linear quasineutral electrostatic drift waves.

Certain kinds of nonlinear instabilities are related to the existence of negative-energy perturbations. In this paper, an exact energy expression for linear quasineutral electrostatic perturbations is derived within the framework of dissipationless multifluid theory that is valid for any geometry. Taking the mass formally as a tensor with, in general, different masses parallel and perpendicular to an ambient magnetic field allows one to treat in a convenient way different approximations such as the full dynamics and restriction to parallel dynamics or the completely adiabatic case. Application to slab configurations yields the result that the adiabatic approximation does not allow negative energy for perturbations which are perfectly localized at a mode resonant surface, whereas inclusion of the parallel dynamics does. This is in agreement with a recent numerical study of drift-wave turbulence within the framework of collisional two-fluid theory by B. Scott [Phys. Rev. Lett. 65, 3289 (1990); Phys. Fluids B 4, 2468 (1992)]. A dissipationless theory can be formulated in terms of a Lagrangian, from which the energy is immediately obtained. We start with the nonlinear theory. The theory is formulated via a Lagrangian which is written in terms of displacement vectors ${\ensuremath{\xi}}_{\ensuremath{\nu}}$(x,t) such that all constraints are taken into account. The nonlinear energy is obtained from the Lagrangian by standard methods. The procedure used is the same as that developed in a forthcoming paper by Pfirsch and Sudan [Phys. Fluids B (to be published)] for ideal nonlinear magnetohydrodynamics theory. From the exact Lagrangian one obtains the Lagrangian of the linearized theory by simple expansion to second order in ${\ensuremath{\xi}}_{\ensuremath{\nu}}$. This Lagrangian then yields the energy of the linearized theory. People working in this field have hitherto considered only (positive semidefinite) expressions for the nonlinear energy which do not immediately allow one to investigate the existence of linear negative-energy perturbations relating to nonlinear instability.

Lagrangian dynamics of a charged particle in a tokamak magnetic field

The Euler–Lagrange equations for a charged particle in a tokamak magnetic field, in the limit of large aspect ratio, are obtained in terms of toroidal coordinates (r,θ,φ,ṙ,θ̇,φ̇) from the exact Lagrangian, which is expressed in terms of the toroidal and poloidal magnetic fluxes. It differs fundamentally from the standard guiding-center approach, where the local magnetic-field direction defines a spatial axis, and where there is no direct link between the velocity coordinates and the time derivatives of the spatial coordinates, so that application to the particular magnetic geometry of the tokamak is often difficult. In contrast, the tokamak magnetic field is specified at the outset in the present approach. For example, cyclotron motion in the combined poloidal and toroidal fields appears as a libration in real space, superimposed on motion composed of linear nonoscillatory and averaged quadratic oscillatory terms. The primary application that is considered concerns the plasma response to high-frequency waves, using both cold plasma and kinetic treatments. A kinetic expression for cyclotron resonance is obtained that agrees with the gyrokinetic result [see, for example, C. N. Lashmore-Davies and R. O. Dendy, Phys. Fluids B 1, 1565 (1989) and references therein], which differs from the standard expression.

Recursive quadratic programming algorithm that uses an exact augmented Lagrangian function

An algorithm for nonlinear programming problems with equality constraints is presented which is globally and superlinearly convergent. The algorithm employs a recursive quadratic programming scheme to obtain a search direction and uses a differentiable exact augmented Lagrangian as line search function to determine the steplength along this direction. It incorporates an automatic adjustment rule for the selection of the penalty parameter and avoids the need to evaluate second-order derivatives of the problem functions. Some numerical results are reported.

A functional approach to the many-body nuclear problem: the MEC

We show how a functional approach to the nuclear many-body problem can provide a unified scheme to deal with the many facets of the problem and to establish well-defined procedures of approximation. We consider, in particular, the perturbative and the mean-field (loop expansion) approximations to an exact effective bosonic Lagrangian, discussing how well-known computational schemes (hole-line expansion, RPA, etc.) may be obtained as subsets of definite orders in these expansions. An application to the calculation of the longitudinal electromagnetic response function, with the simpleNπ interaction, is finally presented. We show how the pion dynamic and the short-range correlations cooperate to obtain a better agreement with experimental data.

A dynamic theory of transverse freezing in the Sherrington-Kirkpatrick Ising model

The effect of a transverse field Gamma on spin-glass freezing in the Sherrington-Kirkpatrick version of the quantum Ising model is studied by using the thermo-field-dynamic (TFD) method (the real-time finite-temperature quantum-field theory) due to Takahashi and Umezawa as a substitute for the 'n-replica trick'. An exact effective Lagrangian for the average dynamic causal and response functions is derived by using the functional-integral approach. Furthermore, the properties of the Lagrangian are studied in the mean-field limit by considering the infinite-range random exchange. In this limit the self-consistent equations for the matrix components of the TFD causal Green function are derived with the effective thermal Hamiltonian containing the dynamic spin self-interaction. The static properties of this equation are studied above and below the freezing temperature Tc by extracting the time-delocalised parts of dynamic autocorrelations. The self-consistent equations for the spin-glass order parameter and local susceptibility are obtained. The classical ( Gamma to 0) and quantum (T to 0) limits are discussed. The crucial role of the choice of the thermal-state condition for the spin-glass problem and its striking similarity to the non-equilibrium formulation of the TFD are pointed out.

New results on a class of exact augmented Lagrangians

In this paper, a new continuously differentiable exact augmented Lagrangian is introduced for the solution of nonlinear programming problems with compact feasible set. The distinguishing features of this augmented Lagrangian are that it is radially unbounded with respect to the multiplier and that it goes to infinity on the boundary of a compact set containing the feasible region. This allows one to establish a complete equivalence between the unconstrained minimization of the augmented Lagrangian on the product space of problem variables and multipliers and the solution of the constrained problem.

An Exact Augmented Lagrangian Approach to Multilevel Optimization

An approach based on the exact augmented Lagrangian function is developed for the optimization of large scale systems constituted by interconnected units. The decomposition and coordination strategies are examined and various schemes of upper level coordination are proposed, all of which are formulated as unconstrained quadratic minimization problems. Convergence analysis is performed exploiting a parallel with minimization by relaxation methods. A numerical example is included.

Exact augmented lagrangian approach to multilevel optimization of large-scale systems

An approach based on the exact augmented lagrangian function is developed for the optimization of large-scale systems composed of interconnected units. The decomposition and coordination strategies are examined and various schemes of upper-level coordination are proposed, all of which are formulated as unconstrained quadratic minimization problems. Convergence analysis is performed exploiting an analogy with minimization by relaxation methods. Numerical examples are reported.

Lagrangian embeddings and critical point theory

We derive a lower bound for the number of intersection points of an exact Lagrangian embedding of a compact manifold into its cotangent bundle with the zero section. To do this the intersection problem is converted into the problem of finding solutions of a Hamiltonian system satisfying canonical boundary conditions. The dynamical problem is then solved by global variational methods on a Hilbert manifold.