Lagrange Structure Research Articles

A new continuum model for general relativistic viscous heat-conducting media.

The lack of formulation of macroscopic equations for irreversible dynamics of viscous heat-conducting media compatible with the causality principle of Einstein's special relativity and the Euler-Lagrange structure of general relativity is a long-lasting problem. In this paper, we propose a possible solution to this problem in the framework of SHTC equations. The approach does not rely on postulates of equilibrium irreversible thermodynamics but treats irreversible processes from the non-equilibrium point of view. Thus, each transfer process is characterized by a characteristic velocity of perturbation propagation in the non-equilibrium state, as well as by an intrinsic time/length scale of the dissipative dynamics. The resulting system of governing equations is formulated as a first-order system of hyperbolic equations with relaxation-type irreversible terms. Via a formal asymptotic analysis, we demonstrate that classical transport coefficients such as viscosity, heat conductivity, etc., are recovered in leading terms of our theory as effective transport coefficients. Some numerical examples are presented in order to demonstrate the viability of the approach. This article is part of the theme issue 'Fundamental aspects of nonequilibrium thermodynamics'.

Connections on the Total Space of a Holomorphic Lie Algebroid

The main purpose of the paper is the study of the total space of a holomorphic Lie algebroid E. The paper is structured in three parts. In the first section, we briefly introduce basic notions on holomorphic Lie algebroids. The local expressions are written and the complexified holomorphic bundle is introduced. The second section presents two approaches on the study of the geometry of the complex manifold E. The first part contains the study of the tangent bundle $$T_{\mathbb {C}}E=T'E\oplus T''E$$ and its link, via the tangent anchor map, with the complexified tangent bundle $$T_{\mathbb {C}}(T'M)=T'(T'M)\oplus T''(T'M)$$ . A holomorphic Lie algebroid structure is emphasized on $$T'E$$ . A special study is made for integral curves of a spray on $$T'E$$ . Theorem 2.8 gives the coefficients of a spray, called canonical, obtained from a complex Lagrangian on $$T'E$$ . In the second part of section two, we study the holomorphic prolongation $$\mathcal {T}'E$$ of the Lie algebroid E. In the third section, we study how a complex Lagrange (Finsler) structure on $$T'M$$ induces a Lagrangian structure on E. Three particular cases are analysed by the rank of the anchor map, the dimensions of manifold M, and those of the fibres. We obtain the correspondent on E of the Chern–Lagrange nonlinear connection from $$T'M$$ .

Extension of the Chern–Simons Theory: Conservation Laws, Lagrange Structures, and Stability

We consider the class of higher derivative 3d vector field models with the wave operator being a polynomial of the Chern–Simons operator. For the nth order theory of this type, we provide a covariant procedure for constructing n-parameter family of conservation laws associated with spatiotemporal symmetries. This family includes the canonical energy that is unbounded from below, whereas others conservation laws from the family can be bounded from below for certain combinations of the Lagrangian parameters, even though higher derivatives are present in the Lagrangian. We prove that any conserved quantity bounded from below is related with invariance of the theory with respect to the time translations and ensures the stability of the model.

Variational tricomplex of a local gauge system, Lagrange structure and weak Poisson bracket

We introduce the concept of a variational tricomplex, which is applicable both to variational and nonvariational gauge systems. Assigning this tricomplex with an appropriate symplectic structure and a Cauchy foliation, we establish a general correspondence between the Lagrangian and Hamiltonian pictures of one and the same (not necessarily variational) dynamics. In practical terms, this correspondence allows one to construct the generating functional of a weak Poisson structure starting from that of a Lagrange structure. As a byproduct, a covariant procedure is proposed for deriving the classical BRST charge of the BFV formalism by a given BV master action. The general approach is illustrated by the examples of Maxwell’s electrodynamics and chiral bosons in two dimensions.

Analytic smoothing effect for a system of Schrödinger equations with three wave interaction

We consider the global Cauchy problem for a system of Schrödinger equations with quadratic interaction. Two types of analytic smoothing effect for the solutions are formulated in the small data setting under the mass resonance condition. One is the usual analytic smoothing effect in space variables in terms of the generator of Galilei transforms. We prove the existence and uniqueness of global solutions which are analytic with respect to Galilei generators for sufficiently small data with exponential decay at infinity in space ℝn with n ≥ 3. The other is analytic smoothing effect in space-time variables in terms of generator of pseudo-conformal and Galilei transforms. We prove the existence and uniqueness of global solutions which are analytic with respect to pseudo-conformal and Galilei generators for sufficiently small data with exponential decay in ℝ4. We also discuss the associated Lagrange structure.

Analytic smoothing effect for a system of Schrödinger equations with two wave interaction

We study the global Cauchy problem for a system of Schrödinger equations with two wave interaction of quadratic, cubic and quintic degrees. For sufficiently small data with exponential decay at infinity we prove the existence and uniqueness of global solutions which are analytic with respect to Galilei and/or pseudo-conformal generators for sufficiently small data with exponential decay at infinity. This paper is a sequel to our paper [22], where three wave interaction is studied. We also discuss the associated Lagrange structure.

Peierls brackets in non-Lagrangian field theory

The concept of Lagrange structure allows one to systematically quantize the Lagrangian and non-Lagrangian dynamics within the path-integral approach. In this paper, I show that any Lagrange structure gives rise to a covariant Poisson bracket on the space of solutions to the classical equations of motion, be they Lagrangian or not. The bracket generalize the well-known Peierls' bracket construction and make a bridge between the path-integral and the deformation quantization of non-Lagrangian dynamics.

BRST analysis of general mechanical systems

We study the groups of local BRST cohomology associated to the general systems of ordinary differential equations, not necessarily Lagrangian or Hamiltonian. Starting with the involutive normal form of the equations, we explicitly compute certain cohomology groups having clear physical meaning. These include the groups of global symmetries, conservation laws and Lagrange structures. It is shown that the space of integrable Lagrange structures is naturally isomorphic to the space of weak Poisson brackets. The last fact allows one to establish a direct link between the path-integral quantization of general not necessarily variational dynamics by means of Lagrange structures and the deformation quantization of weak Poisson brackets.

Passivity-based control of two-dimensional translational oscillator with rotational actuator

An underactuated two-dimensional translational oscillator with rotational actuator (2DTORA) consisting of an actuated rotational proof-mass and two unactuated translational carts is presented. Passivity-based control design is employed for 2DTORA based on its Euler–Lagrange structure and passivity property. Firstly, the dynamics of 2DTORA are derived based on Euler–Lagrange equations. Motivated by constructing a damped close-loop Euler–Lagrange system, the controller dynamics is designed to shape the potential energy and inject the required damping. As a result, the designed controller stabilizes the underactuated 2DTORA with the feedback of the rotational actuator’s position only. Moreover, by modifying controller dynamics with a saturation function, the control input can be constrained to certain bounds. Finally, simulation results demonstrate the feasibility and effectiveness of the proposed controllers.

Parent formulations, frame-like Lagrangians, and generalized auxiliary fields

We elaborate on the recently proposed Lagrangian parent formulation. In particular, we identify a natural choice of the allowed field configurations ensuring the equivalence of the parent and the starting point Lagrangians. We also analyze the structure of the generalized auxiliary fields employed in the parent formulation and establish the relationship between the parent Lagrangian and the recently proposed Lagrange structure for the unfolded dynamics. As an illustration of the parent formalism a systematic derivation of the frame-like Lagrangian for totally symmetric fields starting from the Fronsdal one is given. We also present a concise and manifestly sp(2)-symmetric form of the off-shell constraints and gauge symmetries for AdS higher spin fields at the nonlinear level.

Lagrange Structure and Dynamics for Solutions to the Spherically Symmetric Compressible Navier-Stokes Equations

The compressible Navier-Stokes system (CNS) with density-dependent viscosity coefficients is considered in multi-dimension, the prototype of the system is the viscous Saint-Venat model for the motion of shallow water. A spherically symmetric weak solution to the free boundary value problem for CNS with stress free boundary condition and arbitrarily large data is shown to exist globally in time with the free boundary separating fluids and vacuum and propagating at finite speed as particle path, which is continuous away from the symmetry center. Detailed regularity and Lagrangian structure of this solution have been obtained. In particular, it is shown that the particle path is uniquely defined starting from any non-vacuum region away from the symmetry center, along which vacuum states shall not form in any finite time and the initial regularities of the solution is preserved. Starting from any non-vacuum point at a later-on time, a particle path is also uniquely defined backward in time, which either reaches at some initial non-vacuum point, or stops at a small middle time and connects continuously with vacuum. In addition, the free boundary is shown to expand outward at an algebraic rate in time, and the fluid density decays and tends to zero almost everywhere away from the symmetry center as the time grows up. This finally leads to the formation of vacuum state almost everywhere as the time goes to infinity.

Local BRST cohomologyin(non-)Lagrangian field theory

Some general theorems are established on the local BRST cohomology for not necessarily Lagrangian gauge theories. Particular attention is given to the BRST groups with direct physical interpretation. Among other things, the groups of rigid symmetries and conservation laws are shown to be still connected, though less tightly than in the Lagrangian theory. The connection is provided by the elements of another local BRST cohomology group whose elements are identified with Lagrange structures. This extends the cohomological formulation of the Noether theorem beyond the scope of Lagrangian dynamics. We show that each integrable Lagrange structure gives rise to a Lie bracket in the space of conservation laws, which generalizes the Dickey bracket of conserved currents known in Lagrangian field theory. We study the issues of existence and uniqueness of the local BRST complex associated with a given set of field equations endowed with a compatible Lagrange structure. Contrary to the usual BV formalism, such a complex does not always exist for non-Lagrangian dynamics, and when exists it is by no means unique. The ambiguity and obstructions are controlled by certain cohomology classes, which are all explicitly identified.

On the Lagrangian formulation of multidimensionally consistent systems

Multidimensional consistency has emerged as a key integrability property for partial difference equations (PΔEs) defined on the ‘space–time’ lattice. It has led, among other major insights, to a classification of scalar affine-linear quadrilateral PΔEs possessing this property, leading to the so-called Adler–Bobenko–Suris (ABS) list. Recently, a new variational principle has been proposed that describes the multidimensional consistency in terms of discrete Lagrangian multi-forms. This description is based on a fundamental and highly non-trivial property of Lagrangians for those integrable lattice equations, namely the fact that on the solutions of the corresponding PΔE the Lagrange forms are closed, i.e. they obey a closure relation . Here, we extend those results to the continuous case: it is known that associated with the integrable PΔEs there exist systems of partial differential equations (PDEs), in fact differential equations with regard to the parameters of the lattice as independent variables, which equally possess the property of multidimensional consistency. In this paper, we establish a universal Lagrange structure for affine-linear quad-lattices alongside a universal Lagrange multi-form structure for the corresponding continuous PDEs, and we show that the Lagrange forms possess the closure property.

FINSLER AND LAGRANGE GEOMETRIES IN EINSTEIN AND STRING GRAVITY

We review the current status of Finsler–Lagrange geometry and generalizations. The goal is to aid non-experts on Finsler spaces, but physicists and geometers skilled in general relativity and particle theories, to understand the crucial importance of such geometric methods for applications in modern physics. We also would like to orient mathematicians working in generalized Finsler and Kähler geometry and geometric mechanics how they could perform their results in order to be accepted by the community of "orthodox" physicists. Although the bulk of former models of Finsler–Lagrange spaces where elaborated on tangent bundles, the surprising result advocated in our works is that such locally anisotropic structures can be modeled equivalently on Riemann–Cartan spaces, even as exact solutions in Einstein and/or string gravity, if nonholonomic distributions and moving frames of references are introduced into consideration. We also propose a canonical scheme when geometrical objects on a (pseudo) Riemannian space are nonholonomically deformed into generalized Lagrange, or Finsler, configurations on the same manifold. Such canonical transforms are defined by the coefficients of a prime metric and generate target spaces as Lagrange structures, their models of almost Hermitian/Kähler, or nonholonomic Riemann spaces. Finally, we consider some classes of exact solutions in string and Einstein gravity modeling Lagrange–Finsler structures with solitonic pp-waves and speculate on their physical meaning.

Lagrange structure and quantization

A path-integral quantization method is proposed for dynamical systems whose classical equations of motion do \textit{not} necessarily follow from the action principle. The key new notion behind this quantization scheme is the Lagrange structure which is more general than the Lagrangian formalism in the same sense as Poisson geometry is more general than the symplectic one. The Lagrange structure is shown to admit a natural BRST description which is used to construct an AKSZ-type topological sigma-model. The dynamics of this sigma-model in $d+1$ dimensions, being localized on the boundary, are proved to be equivalent to the original theory in $d$ dimensions. As the topological sigma-model has a well defined action, it is path-integral quantized in the usual way that results in quantization of the original (not necessarily Lagrangian) theory. When the original equations of motion come from the action principle, the standard BV path-integral is explicitly deduced from the proposed quantization scheme. The general quantization scheme is exemplified by several models including the ones whose classical dynamics are not variational.

Structures for implementing FIR-filters with an adjustable frequency response

The paper deals with the problem of implementing FIR-filters with an adjustable amplitude response and a fixed and linear phase response. Among the various possible structures of an FIR-filter, the Lagrange structure, and especially, the frequency sampling structure are taken into consideration because they offer the potential of easy adjustment of the system function at a certain number of sample points in the z-plane. The widely known original forms of these structures are reviewed together with some forms of a new type. The review is followed by a description of several improved forms which, unlike the original forms, do not contain any IIR-subsystems. In addition the close interrelation between the frequency sampling structure and the discrete Fourier transform is outlined. In the final discussion oneor the other of the improved forms is recommended depending on whether the sample points are spaced uniformly or nonuniformly on the unit circle, and also depending on whether the information for adjusting the system function is provided separately from the input signal or it has to be deduced from the input signal by means of a frequency analysis.