Euler Lagrange Expressions Research Articles

The Lagrangian Order-Reduction Theorem in Field Theories

It is known that for every system of variational second order PDEs affine in second derivatives the Tonti Lagrangian is locally reducible to an equivalent first order Lagrangian (Order reduction Theorem). In this paper, a new proof is presented, based on investigation of closed forms related with variational equations, and an explicit formula for the first order Lagrangians arising by order reduction is found. The presented approach extends and completes the Order reduction Theorem by a geometric content and physical meaning of order reducibility: all variational second order PDEs affine in second derivatives admit a first-order covariant Hamiltonian formulation (Hamilton–De Donder equations), i.e. (under certain regularity conditions) carry a multisymplectic structure which is determined directly from the Euler–Lagrange expressions.

Canonical energy and hairy AdS black holes

We propose the modified version of the canonical energy which was introduced originally by Hollands and Wald. Our construction depends only on the Euler-Lagrange expression of the system and thus is independent of the ambiguity in the Lagrangian. After some comments on our construction, we briefly mention on the relevance of our construction to the boundary information metric in the context of the AdS/CFT correspondence. We also study the stability of three-dimensional hairy extremal black holes by using our construction.

A New Systematic Approach to Derive Functionals and Nonclassical Conservation Laws

It is shown in detail that Noether's theorem represents a mathematical identity which always exists for any nondegenerate functional and includes two parts: one is Euler–Lagrange expression, and the other is a possible conservation law. If and only if any physical equations are the same as Euler–Lagrange expressions, the following three points follow: (i) there always exist conservation laws or, at least, balance laws; (ii) the symmetry transformations between inertial frames can be used first to check the absolute invariance; (iii) the field variables included in spacetime transformations can be canceled for absolute invariance, and space and time transformations are irrelevant. A theorem for getting nonclassical conservation laws from the general solution of a physical system is presented. The necessary conditions in a theorem's form are also presented to construct Lagrangians for second- and fourth-order partial differential equations (PDEs). Based on these theorems, the Lagrangians and conservation laws of nonlinear heat and KdV equations are constructed and given. For Boussinesq's solution, a nonclassical conservation integral is also found for application. Moreover, it is shown that there exists only the same conservation laws derived from the generalized variational principle as those of the principle of minimum potential energy in elasticity.

VARIATIONALLY EQUIVALENT PROBLEMS AND VARIATIONS OF NOETHER CURRENTS

We consider systems of local variational problems defining nonvanishing cohomology classes. Symmetry properties of the Euler–Lagrange expressions play a fundamental role since they introduce a cohomology class which adds up to Noether currents; they are related with invariance properties of the first variation, thus with the vanishing of a second variational derivative. In particular, we prove that the conserved current associated with a generalized symmetry, assumed to be also a symmetry of the variation of the corresponding local inverse problem, is variationally equivalent to the variation of the strong Noether current for the corresponding local system of Lagrangians. This current is conserved and a sufficient condition will be identified in order that such a current be global.

Variational principles for natural divergence-free tensors in metric field theories

Let Tab=Tba=0 be a system of differential equations for the components of a metric tensor on Rm. Suppose that Tab transforms tensorially under the action of the diffeomorphism group on metrics and that the covariant divergence of Tab vanishes. We then prove that Tab=Eab(L) is the Euler–Lagrange expression of some Lagrangian density L provided that Tab is of third order. Our result extends the classical works of Cartan, Weyl, Vermeil, Lovelock, and Takens on identifying field equations for the metric tensor with the symmetries and conservation laws of the Einstein equations.

Lovelock tensor as generalized Einstein tensor

We show that the splitting feature of the Einstein tensor, as the first term of the Lovelock tensor, into two parts, namely the Ricci tensor and the term proportional to the curvature scalar, with the trace relation between them is a common feature of any other homogeneous terms in the Lovelock tensor. Motivated by the principle of general invariance, we find that this property can be generalized, with the aid of a generalized trace operator which we define, for any inhomogeneous Euler-Lagrange expression that can be spanned linearly in terms of homogeneous tensors. Then, through an application of this generalized trace operator, we demonstrate that the Lovelock tensor analogizes the mathematical form of the Einstein tensor, hence, it represents a generalized Einstein tensor. Finally, we apply this technique to the scalar Gauss-Bonnet gravity as an another version of string-inspired gravity.

The equivariant inverse problem and the uniqueness of the Yang–Mills equations

The equivariant inverse problem for Yang–Mills-type Euler–Lagrange expressions is solved in the affirmative. This leads to a proof of the uniqueness of the Yang–Mills equations.

The equivariant inverse problem and the Maxwell equations

In affirmative, the equivariant inverse problem for Maxwell-type Euler–Lagrange expressions is solved. This allows the proof of the uniqueness of the Maxwell equations.

On vector-tensor minimally coupled field theories

We consider vector-tensor minimally coupled Lagrangians, i.e., scalar densities of the formℒ = g1/2R +L(g ij ;Ψ i ;Ψ i,j ). We prove that the gauge invariance of any of the sets of Euler-Lagrange expressions implies the gauge invariance of the Lagrangian itself forn even, and an “almost” gauge invariance forn odd. We also find those ℒ for whichE i (ℒ) = 0 orE ij (L) = 0, generalizing well-known results by Lovelock and a result by the authors.

A gravitational Poincaré gauge theory and Higgs mechanism

In this paper we shall construct the Lagrangian of a gravitational Poincaré gauge theory using degeneracy in the Euler–Lagrange expressions as a primary restriction. Such a generalization of a Lorentz gauge theory requires the addition of not only a translation gauge connection, but also a Goldstone field. The intractability of the field equations is lessened somewhat by means of a particular choice of gauge which acts like a Higgs mechanism. With one further assumption a complete reduction to the corresponding Lorentz theory can be made, and the Einstein vacuum field equations with cosmological term are recovered.

The uniqueness of gravity as a Poincaré or Lorentz gauge theory

The uniqueness of the Einstein–Cartan–Sciama–Kibble theory is examined using a concomitant approach. The demand that the Lagrangian be a scalar density under coordinate transformations and a scalar under Poincaré or Lorentz gauge transformations as well as be degenerate in the order of the Euler–Lagrange expressions determines a class of Lagrangians whose Euler–Lagrange equations reduce essentially to the Einstein vacuum field equations with cosmological term.

The principle of minimal gravitational coupling

The principle of minimal gravitational coupling requires that the total Lagrangian for the field equations of general relativity consist of two additive parts, one part corresponding to the free gravitational Lagrangian and the other part to external source fields in curved spacetime. For source fields characterized by scalars, vectors and two-component spinors we prove that this additive decomposition of the total Lagrangian is an inevitable consequence of certain very general assumptions concerning the Euler-Lagrange expressions arising from the total Lagrangian.

Comments on certain divergence-free tensor densities in a 3-space

It is well known that a necessary and sufficient condition for the conformal flatness of a three-dimensional pseudo-Riemannian manifold can be expressed in terms of the vanishing of a third-order tensor density concomitant of the metric which has contravariant valence 2. This was first discovered by Cotton in 1899. It is shown that Cotton’s tensor density is not the Euler–Lagrange expression corresponding to a scalar density built from one metric tensor. This tensor density is shown to be uniquely characterized by its conformal properties coupled with the demand that it be differentiable for arbitrary metrics.

A connection approach to the Einstein-Maxwell field equations

Scalar densities which are concomitants of the metric tensor, a symmetric affine connection and its first derivative, and the derivative of a vector field are examined. Rather simple demands are imposed on the corresponding Euler-Lagrange expressions. It is found that the associated field equations reduce essentially to the Einstein-Maxwell field equations.

Higher Euler operators and some of their applications

A set of operators which are associated with the Euler–Lagrange operator is introduced. An analysis of the commutation properties of these new operators, which will be referred to as the higher Euler operators, leads to a generalization of the necessary conditions for an expression to be an Euler–Lagrange expression. A product rule is derived for the higher Euler operators. In the special case of the Euler–Lagrange operator this product rule is basic to simple proofs of sufficiency theorems for the existence of a Lagrangian given the potential Euler–Lagrange expressions. By considering a certain homogeneity property, a characterization of Lagrangians in terms of their Euler–Lagrange expressions is established. Examples of applications of this characterization are given. A general procedure is given for constructing equivalent (not necessarily scalar density) Lagrangians when the field functions are tensorial and the Euler–Lagrange expressions are tensor densities. These results give particular significance to one of the higher Euler operators.

INVARIANT VARIATIONAL PRINCIPLES INVOLVING VECTOR AND METRIC FIELDS IN 4-DIMENSIONS

Abstract Variational principles in which the Lagrangian is a scalar density and a function of a metric tensor and a vector field, together with their first derivatives, are investigated in a 4-dimensional space. Associated with such Lagrangians are two expressions, the metric Euler-Lagrange expression and the vector Euler-Lagrange expression. The most general Lagrangians (of this kind) for which either of these Euler-Lagrange expressions vanishes identically, are obtained. The most general Lagrangian (of this kind) for which the vector Euler-Lagrange equations are precisely Maxwell's equations is obtained. Although this Lagrangian is more general than the one commonly used, it still has essentially the same energy-momentum tensor. The most general Lagrangian (of this kind) for which the metric Euler-Lagrange expression is precisely the electromagnetic energy-momentum tensor is derived. Although this Lagrangian is also more general than the one commonly used, the associated vector Euler-Lagrange equations ...

Sufficiency conditions under which a lagrangian is an ordinary divergence

An expression is derived for the variation of Lagrangians which are such that the set of admissible variables of variation is star-shaped. If such a Lagrangian leads to identically vanishing Euler-Lagrange expressions then it is shown that under suitable circumstances the Lagrangian in question must be an ordinary divergence. Furthermore, an expression is given for the ‘vector’ field which appears in this ordinary divergence.