Point symmetry group of the Lagrangian

The theory of finite point symmetry transformations is revisited within the frame of the general theory of transformations of Lagrangian mechanics. The point symmetry groupG(L) of a given Lagrangian functionL (i.e., the Noether group) is thus obtained, and its main features are briefly discussed. The explicit calculation of the Noether group is presented for two rather simple c-equivalent Lagrangian systems. The formalism affords an introduction to the Noether theory of infinitesimal point symmetry transformations in Lagrangian mechanics; however, it is also of interest in its own right.     

A new approach to spontaneously broken conformal symmetry

In this paper we present a new method for constructing theories of gravitation which exhibit spontaneously broken conformal symmetry. It does not require introducing nongeometric terms (i.e., auxiliary gauge fields or potential terms for the conformal field) into the Lagrangian. It is based on a theory which initially is locally both Lorentz invariant and Weyl gauge invariant inD dimensions. It is shown that, if the field Lagrangian contains terms quadratic in curvature in addition to the Ricci scalar, then the field equations allow both the dilation field and some connection components to have nonvanishing vacuum values. Both Lorentz and Weyl symmetries are thereby broken simultaneously.     

Two-dimensionalR n-gravitation

A system with constraints is considered: a string theory whose Lagrangian is thenth power of the Gauss curvature of a space-time manifold (n∈N,n>1). The problem is solved exactly because after the constraints are utilized we deal with a variational problem with a trivial Lagrangian, i.e., its Euler-Lagrange equations are satisfied identically. One can say that the constraints “swallow” all dynamical degrees of freedom of the field theory. The investigation is a continuation of the 1989 work of Burlankov and Pavlov, who solved the problem of two-dimensionalR ²-gravitation under the gauge γ=1.     

Superconducting State in the Lagrangian Formalism of the Generalized Hubbard Model

The nonperturbative large-N expansion applied to the generalized Hubbard model describing N-fold-degenerate correlated bands is considered. Our previous results, obtained in the framework of the Lagrangian formalism for the normal-state case, are extended to the superconducting state. The standard Feynman diagrammatics is obtained and the renormalized physical quantities are computed and analyzed. Our purpose is to obtain the 1/N corrections to the renormalized boson and fermion propagators when a state with Cooper-pair condensation (i.e., the superconducting state) is considered.     

Cosmological application on five-dimensional teleparallel theory equivalent to general relativity

A theory of(4+1)-dimensional gravity has been developed on the basis of which equivalent to the theory of general relativity by teleparallel.The fundamental gravitational field variables are the 5-dimensional(5D) vector fields(pentad),defined globally on a manifold M,and gravity is attributed to the torsion.The Lagrangian density is quadratic in the torsion tensor.We then apply the field equations to two different homogenous and isotropic geometric structures which give the same line element,i.e.,FRW in five dimensions.The cosmological parameters are calculated and some cosmological problems are discussed.     

Cosmic acceleration and crossing of in non-minimal modified Gauss–Bonnet gravity

Modified Gauss–Bonnet, i.e., f(G) gravity is a possible explanation of dark energy. Late time cosmology for the f(G) gravity non-minimally coupled with a free massless scalar field have been investigated in Ref. [S. Nojiri, S.D. Odintsov, P.V. Tretyakov, Phys. Lett. B 651 (2007) 224, arXiv:0704.2520 [hep-th]; S. Nojiri, S.D. Odintsov, P.V. Tretyakov, Progr. Theor. Phys. Suppl. 172 (2008) 81, arXiv:0710.5232]. In this Letter we generalize the work of Ref. [S. Nojiri, S.D. Odintsov, P.V. Tretyakov, Phys. Lett. B 651 (2007) 224, arXiv:0704.2520 [hep-th]; S. Nojiri, S.D. Odintsov, P.V. Tretyakov, Progr. Theor. Phys. Suppl. 172 (2008) 81, arXiv:0710.5232] by including scalar potential in the matter Lagrangian which is non-minimally coupled with modified Gauss–Bonnet gravity. Also we obtain the conditions for having a much more amazing problem than the acceleration of the universe, i.e. crossing of ω=−1, in f(G) non-minimally coupled with tachyonic Lagrangian.     

Nuclear mean field from chirally symmetric effective theory

The mean-field theory of the nuclear many-body problem proposed recently by Furnstahl, Serot, and Tang (FST) is discussed. The FST chiral Lagrangian is derived in terms of an effective field theory. This new approach allows one to construct in a controlled manner the universal nuclear Lagrangian consistent with symmetries of QCD. The FST Lagrangian is constructed by using power counting, i.e., the expansion in powers of the lowest lying hadronic fields and their derivatives. Terms in the Lagrangian are organized by applying Georgi’s naive dimensional analysis and “naturalness” condition. The relevant degrees of freedom are nucleons, pions, an isoscalar-vector field ω meson), an isoscalar-scalar field (σ meson), and an isovector-vector field (ρ meson). The chiral symmetry is realized nonlinearly using a standard WCCWZ procedure.     

Metric-affine variational principles in general relativity II. Relaxation of the Riemannian constraint

We continue our investigation of a variational principle for general relativity in which the metric tensor and the (asymmetric) linear connection are varied independently. As in Part I, the matter Lagrangian is minimally coupled to the connection and the gravitational Lagrangian is taken to be the curvature scalar, but we now relax the Riemannian constraint as far as possible—that is, as far as the projective invariance of the assumed gravitational Lagrangian will allow. The outcome of this procedure is a gravitational theory formulated in a volume-preserving space-time (i.e., with torsion and tracefree nonmetricity). The vanishing of the trace of the nonmetricity is due to the remaining vector constraint. We also discuss the physical significance of the relaxation of the Riemannian constraint, the possible relaxation of the vector constraint, the notion of the hypermomentum current, and its possible relation to elementary particle physics.     

Derivation of Nonlinear Schrödinger Equation

We propose some nonlinear Schrödinger equations by adding some higher order terms to the Lagrangian density of Schrödinger field, and obtain the Gross-Pitaevskii (GP) equation and the logarithmic form equation naturally. In addition, we prove the coefficient of nonlinear term is very small, i.e., the nonlinearity of Schrödinger equation is weak.     

Vibration of complex structures: The modal constraint method

A method is described for establishing the natural frequencies of an arbitrary structure with arbitrary supports. The method is based on the modal constraint technique described in a previous paper [1]. As shown in the present paper Weinstein's theory for the intermediate problem can be regarded as equivalent to the Lagrangian multiplier method: i.e., both methods result in the same eigenvalue equations. Weinstein's theory deals with modifications of base differential operators whereas the Lagrangian multiplier method deals with modifications of base energy functionals. The modal constraint technique is an extension of Weinstein's theory, or in energy terms the generalized Fourier expansion of the Lagrangian multiplier. The merits of this method lie in the fact that the eigenvalues and eigenfunctions of structures are used as base structures. The coupling of these structures are taken into account by Lagrangian generalized forces of the constraint acting on the base structures. Some examples are given and the results compared with known solutions.     

Chaotic particle motion under linear surface waves

We investigate the motion of infinitesimal particles in the flow field inside the fluid under a traveling surface wave. It is shown that, even for two-dimensional waves, a superposition of two or more traveling harmonic waves is enough to generate chaotic particle motion, i.e., Lagrangian chaos. (c) 1996 American Institute of Physics.     

Statistical Properties of the Burgers Equation with Brownian Initial Velocity

We study the one-dimensional Burgers equation in the inviscid limit for Brownian initial velocity (i.e. the initial velocity is a two-sided Brownian motion that starts from the origin x=0). We obtain the one-point distribution of the velocity field in closed analytical form. In the limit where we are far from the origin, we also obtain the two-point and higher-order distributions. We show how they factorize and recover the statistical invariance through translations for the distributions of velocity increments and Lagrangian increments. We also derive the velocity structure functions and we recover the bifractality of the inverse Lagrangian map. Then, for the case where the initial density is uniform, we obtain the distribution of the density field and its n-point correlations. In the same limit, we derive the n-point distributions of the Lagrangian displacement field and the properties of shocks. We note that both the stable-clustering ansatz and the Press-Schechter mass function, that are widely used in the cosmological context, happen to be exact for this one-dimensional version of the adhesion model.     

Practical use of variational principles for modeling water waves

This paper describes a method for deriving approximate equations for irrotational water waves. The method is based on a ‘relaxed’ variational principle, i.e., on a Lagrangian involving as many variables as possible. This formulation is particularly suitable for the construction of approximate water wave models, since it allows more freedom while preserving a variational structure. The advantages of this relaxed formulation are illustrated with various examples in shallow and deep waters, as well as arbitrary depths. Using subordinate constraints (e.g., irrotationality or free surface impermeability) in various combinations, several model equations are derived, some being well-known, other being new. The models obtained are studied analytically and exact traveling wave solutions are constructed when possible.     

An Analytical Approach to the Turbulence-Relaxed State in Tokamak Plasmas

Starting from the continuity, temperature, and motion equations of the trapped electron fluid in generaltokamak magnetic field with positive or reversed shear and the definition of Lagrangian invariant, dL / dt = ( t u. )L =0, where u is convective velocity, the trapped electron dynamics is considered in the following two assumptions: (i) theturbulence is low frequency electrostatic, and (ii) L is a functional only of the density n, temperature T, and magneticfield B, and the effect of perturbation potential φ is included in the convective velocity u, i.e., u is a functional of n,T, B, and φ. The Lagrangian invariant hidden in the trapped electron dynamics is strictly found: L= ln[(n/B)c1(T/B2/3)c2], where c1 and c2 are dimensionless changeable parameters and c1 ∝ c2. From this Lagrangian invariant thewhich, in the limit of large aspect ratio, reduce to n(r)q(r) = const. and T3/2(r)q(r) = const., respectively. The lattertwo scaling laws are compared with existent experimental results, being in good agreement.     

Electromagnetic field with induced massive term: Case with scalar field

We consider an interacting system of massless scalar and electromagnetic fields, with the Lagrangian explicitly depending on the electromagnetic potentials, i.e., interaction with broken gauge invariance. The Lagrangian for interaction is chosen in such a way that the electromagnetic field equation acquires an additional term, which in some cases is proportional to the vector potential of the electromagnetic field. This equation can be interpreted as the equation of motion of photon with induced nonzero rest-mass. This system of interacting fields is considered within the scope of Bianchi type-I (BI) cosmological model. It is shown that, as a result of interaction the isotropization process of the expansion takes place.     

Twistor connection and the Palatini method

For the Yang-Mills Lagrangian of the twistor connection, an analog of the Palatini variational method is considered in which the variations of the twistor connection A_m and metric g_ab, are taken to be independent. It is shown that varying the Lagrangian with respect to the connection establishes a relation between A_m and g_ab (i.e., defines a standard twistor connection, postulated earlier), while varying with respect to the metric with a subsequent substitution of explicit expressions of the standard twistor connection leads to the Bach vacuum equations, describing the dynamics of conformal gravity.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 94–97, May, 1988.The author thanks Professor N. V. Mitskevich for his attention to this work and useful discussions.     

Structure of Shocks in Burgers Turbulence¶with Stable Noise Initial Data

Burgers equation can be used as a simplified model for hydrodynamic turbulence. The purpose of this paper is to study the structure of the shocks for the inviscid equation in dimension 1 when the initial velocity is given by a stable Lévy noise with index α∈ (1/2,2]. We prove that Lagrangian regular points exist (i.e. there are fluid particles that have not participated in shocks at any time between 0 and t) if and only if α≤ 1 and the noise is not completely asymmetric, and that otherwise the shock structure is discrete. Moreover, in the Cauchy case α= 1, we show that there are no rarefaction intervals, i.e. at time t >0$, there are fluid particles in any non-empty open interval. Received: 28 September 1998 / Accepted: 12 January 1999     

Supersymmetry parameter analysis: SPA convention and project

High-precision analyses of supersymmetry parameters aim at reconstructing the fundamental supersymmetric theory and its breaking mechanism. A well defined theoretical framework is needed when higher-order corrections are included. We propose such a scheme, Supersymmetry Parameter Analysis SPA, based on a consistent set of conventions and input parameters. A repository for computer programs is provided which connect parameters in different schemes and relate the Lagrangian parameters to physical observables at LHC and high energy e ⁺ e^- linear collider experiments, i.e., masses, mixings, decay widths and production cross sections for supersymmetric particles. In addition, programs for calculating high-precision low energy observables, the density of cold dark matter (CDM) in the universe as well as the cross sections for CDM search experiments are included. The SPA scheme still requires extended efforts on both the theoretical and experimental side before data can be evaluated in the future at the level of the desired precision. We take here an initial step of testing the SPA scheme by applying the techniques involved to a specific supersymmetry reference point.     

An Analytical APproach to the Turbulence—Relaxed State in Tokamak Plasmas

Starting from the continuity, temperature, and motion equations of the trapped electron fluid in general tokamak magnetic field with positive or reversed shear and the definition of Lagrangian invariant, dL/dt≡(\partial_t+ u•▽)L=0, where u is convective velocity, the trapped electron dynamics is considered in the following two assumptions: (i) the turbulence is low frequency electrostatic, and (ii) L is a functional only of the density n, temperature T, and magnetic field B, and the effect of perturbation potential φ is included in the convective velocity u, i.e., u is a functional of n, T, B, and φ. The Lagrangian invariant hidden in the trapped electron dynamics is strictly found: L=ln[(n/B)^c₁(T/B^2/3)^c₂], where c₁ and c₂ are dimensionless changeable parameters and c₁∝c₂. From this Lagrangian invariant the turbulent particle and electron thermal transport scaling laws are derived: 〈n>_ψq(ψ)=const. and 〈T^3/2>_ψq(ψ)=const., which, in the limit of large aspect ratio, reduce to n(r)q(r)=const. and T^3/2(r)q(r)=const., respectively. The latter two scaling laws are compared with existent experimental results, being in good agreement.     

ELLAM for resolving the kinematics of two-dimensional resistive magnetohydrodynamic flows

We combine the finite element method with the Eulerian–Lagrangian Localized Adjoint Method (ELLAM) to solve the convection–diffusion equations that describe the kinematics of magnetohydrodynamic flows, i.e., the advection and diffusion of a magnetic field. Simulations of three two-dimensional test problems are presented and in each case we analyze the energy of the magnetic field as it evolves towards its equilibrium state. Our numerical results highlight the accuracy and efficiency of the ELLAM approach for convection-dominated problems.