Null Lagrangian Research Articles

ANALISIS LAGRANGIAN NULL NONSTANDAR DAN FUNGSI GAUGE UNTUK HUKUM INERSIA NEWTON : SEBUAH REVIEW

This paper describes a review of a journal entitled 'Nonstandard Null Lagrangians and Gauge Functions for Newtonian Law of Inertia' which discusses Nonstandard Lagrangian Null solutions for Newton's Law of Inertia. The purpose of this study is to present in detail the Lagrangian Formalism method for generating Nonstandard Lagrangian Null and its Gauge function for Newton's Law of Inertia, as well as the role of action invariant in generating Lagrangian Null and Exact Gauge functions, by deriving a one-dimensional oscillator arm using the basic Lagrangian equations. The Nonstandard Null Lagrangian is derived from the Nonstandard Lagrangian, then the two Lagrangian Null are entered into the action invariant to make it Exact, after the Nonstandard Lagrangian Null is declared Exact, it is substituted into which represents Newton's Law of Inertia. The results of this research show that an Exact Nonstandard Lagrangian Null can be generated by making the Gauge Function Invariant, so that the first Nonstandard Lagrangian Null and Gauge Function for Newton's Law of Inertia are obtained.

Nonstandard Null Lagrangians and Gauge Functions for Newtonian Law of Inertia

New null Lagrangians and gauge functions are derived and they are called nonstandard because their forms are different than those previously found. The invariance of the action is used to make the Lagrangians and gauge functions exact. The first exact nonstandard null Lagrangian and its gauge function for the law of inertia are obtained, and their physical implications are discussed.

Null Lagrangians in Cosserat Elasticity

In the framework of nonlinear theory of Cosserat elasticity, also called micropolar elasticity, we provide the complete characterization of null Lagrangians for three dimensional bodies as well as for shells. Using the Gibb’s rotation vector for description of the microrotation, this task is possible by an application of a theorem stated by Olver and Sivaloganathan in ‘the structure of null Lagrangians’ (Nonlinearity 1:389–398, 1988). A set of necessary and sufficient conditions is also provided for the elasticity tensors to correspond to a null Lagrangian in linearized micropolar theory.

Null Lagrangians in linear theories of micropolar type and few other generalizations of elasticity

In the context of linear theories of generalized elasticity including those for homogeneous micropolar media, quasicrystals, piezoelectric and piezomagnetic media, we explore the concept of null Lagrangians. For obtaining the family of null Lagrangians we employ the sufficient conditions of H. Rund. In some cases a non-zero null Lagrangian is found and the stored energy admits a split into a null Lagrangian and a remainder. However, the null Lagrangian vanishes whenever the relevant elasticity tensor obeys certain symmetry conditions which can be construed as an analogue of the Cauchy relations.

On Null Lagrangians

Abstract We consider Lagrangians for parametric variational problems defined on velocity manifolds and show that a Lagrangian is null precisely when its shadow, a family of vector forms, is closed. We also show that a null Lagrangian can be recovered (to within a constant) from its shadow, and therefore that such a Lagrangian is (again to within a constant) a sum of determinants of total derivatives.

Solutions of the differential inequality with a null Lagrangian: higher integrability and removability of singularities. II

The aim of this paper is to establish a result on removability of singularities for solutions of the differential inequality with a null Lagrangian. Also, we obtain integral estimates for wedge products of closed differential forms and for minors of a Jacobian matrix.

Solutions of the differential inequality with a null Lagrangian: higher integrability and removability of singularities. I

The aim of this paper is to derive the self-improving property of integrability for derivatives of solutions of the differential inequality with a null Lagrangian. More precisely, we prove that the solution of the Sobolev class with some Sobolev exponent slightly smaller than the natural one determined by the structural assumption on the involved null Lagrangian actually belongs to the Sobolev class with some Sobolev exponent slightly larger than this natural exponent. We also apply this property to improve Holder regularity and stability theorems of [19].

Sequential weak continuity of null Lagrangians at the boundary

We show weak* in measures on \(\bar{\Omega }\)/ weak-\(L^1\) sequential continuity of \(u\mapsto f(x,\nabla u):W^{1,p}(\Omega ;\mathbb{R }^m)\rightarrow L^1(\Omega )\), where \(f(x,\cdot )\) is a null Lagrangian for \(x\in \Omega \), it is a null Lagrangian at the boundary for \(x\in \partial \Omega \) and \(|f(x,A)|\le C(1+|A|^p)\). We also give a precise characterization of null Lagrangians at the boundary in arbitrary dimensions. Our results explain, for instance, why \(u\mapsto \det \nabla u:W^{1,n}(\Omega ;\mathbb{R }^n)\rightarrow L^1(\Omega )\) fails to be weakly continuous. Further, we state a new weak lower semicontinuity theorem for integrands depending on null Lagrangians at the boundary. The paper closes with an example indicating that a well-known result on higher integrability of determinant by Muller (Bull. Am. Math. Soc. New Ser. 21(2): 245–248, 1989) need not necessarily extend to our setting. The notion of quasiconvexity at the boundary due to J.M. Ball and J. Marsden is central to our analysis.

Forms of null Lagrangians in field theories of continuum mechanics

The divergence representation of a null Lagrangian that is regular in a star-shaped domain is used to obtain its general expression containing field gradients of order ≤ 1 in the case of spacetime of arbitrary dimension. It is shown that for a static three-component field in the three-dimensional space, a null Lagrangian can contain up to 15 independent elements in total. The general form of a null Lagrangian in the four-dimensional Minkowski spacetime is obtained (the number of physical field variables is assumed arbitrary). A complete theory of the null Lagrangian for the n-dimensional spacetime manifold (including the four-dimensional Minkowski spacetime as a special case) is given. Null Lagrangians are then used as a basis for solving an important variational problem of an integrating factor. This problem involves searching for factors that depend on the spacetime variables, field variables, and their gradients and, for a given system of partial differential equations, ensure the equality between the scalar product of a vector multiplier by the system vector and some divergence expression for arbitrary field variables and, hence, allow one to formulate a divergence conservation law on solutions to the system.

Sharp bounds on the volume fractions of two materials in a two-dimensional body from electrical boundary measurements: the translation method

We deal with the problem of estimating the volume of inclusions using a small number of boundary measurements in electrical impedance tomography. We derive upper and lower bounds on the volume fractions of inclusions, or more generally two phase mixtures, using two boundary measurements in two dimensions. These bounds are optimal in the sense that they are attained by certain configurations with some boundary data. We derive the bounds using the translation method which uses classical variational principles with a null Lagrangian. We then obtain necessary conditions for the bounds to be attained and prove that these bounds are attained by inclusions inside which the field is uniform. When special boundary conditions are imposed the bounds reduce to those obtained by Milton and these in turn are shown here to reduce to those of Capdeboscq–Vogelius in the limit when the volume fraction tends to zero. The bounds of this article, and those of Milton, work for inclusions of arbitrary volume fractions. We then perform some numerical experiments to demonstrate how good these bounds are.

Sufficient conditions for the validity of the Cauchy-Born rule close to SO(n)

The Cauchy–Born rule provides a crucial link between continuum theories of elasticity and the atomistic nature of matter. In its strongest form it says that application of affine displacement boundary conditions to a monatomic crystal will lead to an affine deformation of the whole crystal lattice. We give a general condition in arbitrary dimensions which ensures the validity of the Cauchy–Born rule for boundary deformations which are close to rigid motions. This generalizes results of Friesecke and Theil [J. Nonlin. Sci. 12 (2002), 445–478] for a two-dimensional model. As in their work the key idea is to use a discrete version of polyconvexity (ordinary convexity of the elastic energy as a function of the atomic positions is ruled out by frame-indifference). The main point is the construction of a suitable discrete null Lagrangian which allows one to separate rigid motions. To do so we observe a simple identity for the determinant function on \operatorname{SO}(n) and use interpolation to convert ordinary null Lagrangians into discrete ones.

On null Lagrangians

We consider multiple-integral variational problems where the Lagrangian function, defined on a frame bundle, is homogeneous. We construct, on the corresponding sphere bundle, a canonical Lagrangian form with the property that it is closed exactly when the Lagrangian is null. We also provide a straightforward characterization of null Lagrangians as sums of determinants of total derivatives. We describe the correspondence between Lagrangians on frame bundles and those on jet bundles: under this correspondence, the canonical Lagrangian form becomes the fundamental Lepage equivalent. We also use this correspondence to show that, for a single-determinant null Lagrangian, the fundamental Lepage equivalent and the Carathéodory form are identical.

Equivalent and separable strain-energy functions in compressible, finite elasticity

A material non-uniqueness is identified for isotropic, homogeneous, hyperelastic, compressible materials. This non-uniqueness is a special case of the hyperelastic null Lagrangian. Equivalence classes of strain-energy functions (SEFs) are then obtained. This equivalence can be used to extend in a natural way some of the common SEF used in incompressible elasticity to compressible elasticity. The familiar Valanis–Landel separable form of the SEF for incompressible materials is extended in this way. Some necessary restrictions are imposed on this new form and its behaviour in uniaxial tension is discussed. Previous problems in using the Valanis–Landel form of the SEF in compressible elasticity have been overcome.

Gaussian curvature as a null homogeneous second-order Lagrangian

Gaussian curvature as a null homogeneous second-order Lagrangian

Nonlinear dynamics of the classical isotropic Heisenberg antiferromagnetic chain: The sigma model sector and the kink sector

We identify two distinct low-energy sectors in the classical isotropic antiferromagnetic Heisenberg spin-S chain. In the continuum limit, we show that two types of rotation generators arise for the field in each sector. Using these, the Lagrangian for sector I is shown to be that of the nonlinear sigma model. Sector II has a null Lagrangian; Its Hamiltonian density is just the Pontryagin term. Exact solutions are found in the form of magnons and precessing pulses in I and moving kinks in II. The kink has `spin' S. Sector I has a higher minimum energy than II.

NULL LAGRANGIANS IN LINEAR ELASTICITY

The concept of null Lagrangian is exploited in the context of linear elasticity. In particular, it is shown that the stored energy functional can always be split into a null Lagrangian and a remainder; the null Lagrangian vanishes if and only if the elasticity tensor obeys the Cauchy relations, and is therefore determined by only 15 independent moduli (the so-called “rari-constant” theory of elasticity).

Second variation of liquid crystal energy at x /| x |

A new and simplified proof of the sign of the second variation of the Oseen–Frank energy in terms of the elastic constants is given. The proof relies on the frame indifference of the energy and a new expression for the second invariant null lagrangian as a surface integral.

The structure of null Lagrangians

It is said that L(x,u, Del u) is a null Lagrangian if and only if the corresponding integral functional E(u)= integral Omega L(x,u, Del u) dx has the property that E(u+ phi )=E(u) For all phi in C0infinity ( Omega ), for any choice of u in C1( Omega ). In the homogeneous case, corresponding to L(x,u, Del u)= Phi ( Del u), it is known that a necessary and sufficient condition for L to be a null Lagrangian is that Phi ( Del u) is an affine combination of subdeterminants of Del u of all orders. The authors show that all inhomogeneous null Lagrangians may be constructed from these homogeneous ones by introducing appropriate potentials.

Null Lagrangians, admissible tractions, and finite element methods

The principle of virtual work in elastostatics is shown to imply the principle of minimum potential energy whenever the applied tractions obtain as natural Neumann data associated with a null Lagrangian. Tractions of this kind are referred to as variationally admissible, and are those for which finite element methods may be applied directly. The most general null Lagrangian for elastostatic problems is shown to be generated by 15 functions of the independent variables and the displacements (current configuration variables), and this representation remains valid both for linear and nonlinear elasticity. Explicit representations are given for all null Lagrangians and all variationally admissible tractions, the general form of a null Lagrangian being a third-degree polynomial in the configuration gradients. Explicit application to finite element formulations of elastostatic problems is given. A model problem of a Sturm-Liouville system with nonlinear Neumann data at one end is given in the appendix as an assist to the reader.

Toward a self-consistent theory of elastoelectromagnetic neumann boundary value problems

The class of null Lagrangians is used to model inhomogeneous Neumann boundary data for elastic bodies that interact with electromagnetic fields in the presence of defects (dislocations and disclinations). The interface conditions of the various fields on the common boundary of two such bodies are studied, and the relations between the derivatives of null Lagrangians and corresponding quantities defined on the interface are derived. Explicit evaluations of the interface conditions, for a specific selected null Lagrangian, are given for an elastoelectromagnetic material with and without defects. Loading devices for the application of boundary tractions are also defined for elastic materials.