Euler-Lagrange Operator Research Articles

New conservation laws of the Boussinesq and generalized Kadomtsev–Petviashvili equations via homotopy operator

According to the tools of linear algebra and calculus of variations, the conservation laws of Boussinesq and generalized Kadomtsev–Petviashvili (gKP) equations are investigated using multipliers and scaling methods. Using the Euler–Lagrange operator, the determining equations are calculated to find the multipliers of Boussinesq equation and the actual density of gKP equation. Then, the flux and density pairs of Boussinesq equation and the corresponding flux of gKP equation are obtained through 2-dimensional homotopy operator.

New role of null lagrangians in derivation of equations of motion for dynamical systems

The space of null Lagrangians is the least investigated territory in dynamics as these Lagrangians are identically sent to zero by their Euler–Lagrange operator, and thereby they are having no effects on equations of motion. A procedure that significantly generalizes the previous work, which appeared in (2022, Physica Scripta 97, 125213), is developed and used to construct null Lagrangians and then the corresponding non-standard Lagrangians, which represent a range of interesting dynamical systems. By using the generalized procedure, derivation of equations of motion for a harmonic oscillator as well as for the Bateman and Duffing oscillators is presented. The obtained results demonstrate a new role played by the null Lagrangians and their corresponding non-standard Lagrangians in describing linear and nonlinear, and dissipative and non-dissipative dynamical systems.

The Schwarzian derivative and Euler–Lagrange equations

We study the Schwarzian derivative from a variational viewpoint. In particular, we show that the Schwarzian derivative defines a first integral of the Euler–Lagrange equation of a second order Lagrangian. Moreover, we prove that the Schwarzian derivative itself can be considered as the Euler–Lagrange operator for an appropriately chosen class of variations.

On the spectrum of Euler–Lagrange operator in the stability analysis of Bénard problem

In studying the stability of Bénard problem, we usually have to solve a variational problem to determine the critical Rayleigh number for linear or nonlinear stability. To solve the variational problem, one usually transforms it to an eigenvalue problem which is called Euler–Lagrange equations. An operator related to the Euler–Lagrange equations is usually referred to as Euler–Lagrange operator whose spectrum is investigated in this paper. We have shown that the operator possesses only the point spectrum consisting of real number, which forms a countable set. Moreover, it is found that the spectrum of the Euler–Lagrange operator depends on the thickness of the fluid layer.

The interplay between two Euler\u2013Lagrange operators relating to the nonlinear elliptic system Sigma [(u, {mathscr {P}}), varOmega

We establish the existence of multiple whirling solutions to a class of nonlinear elliptic systems in variational form subject to pointwise gradient constraint and pure Dirichlet type boundary conditions. A reduced system for certain mathbf{SO}(n)-valued matrix fields, a description of its solutions via Lie exponentials, a structure theorem for multi-dimensional curl free vector fields and a remarkable explicit relation between two Euler–Lagrange operators of constrained and unconstrained types are the underlying tools and ideas in proving the main result.

On the initial boundary value problem of a Navier-Stokes/$Q$-tensor model for liquid crystals

This work is concerned with the solvability of a Navier-Stokes/Q-tensor coupled system modeling the nematic liquid crystal flow on a bounded domain in three dimensional Euclidian space with strong anchoring boundary condition for the order parameter. We prove the existence and uniqueness of local in time strong solutions to the system with an anisotropic elastic energy. The proof is based on mainly two ingredients: first, we show that the Euler-Lagrange operator corresponding to the Landau-de Gennes free energy with general elastic coefficients fulfills the strong Legendre condition. This result together with a higher order energy estimate leads to the well-posedness of the linearized system, and then a local in time solution of the original system which is regular in temporal variable follows via a fixed point argument. Secondly, the hydrodynamic part of the coupled system can be reformulated into a quasi-stationary Stokes type equation to which the regularity theory of the generalized Stokes system, and then a bootstrap argument can be applied to enhance the spatial regularity of the local in time solution.

Principal symbol of Euler–Lagrange operators

We shall introduce the principal symbol for quite a general class of (quasi linear) Euler–Lagrange operators and use them to characterise well-posed initial value problems in gauge covariant field theories. We shall clarify how constraints can arise in covariant Lagrangian theories by extending the standard treatment in GR and without resorting to Hamiltonian formalism. Finally as an example of application, we sketch a quantisation procedure based on what is done in LQG by framing it in a more general context which applies to general gauge covariant field theories.

Finite time singularities for the locally constrained Willmore flow of surfaces

In this paper we study the steepest descent $L^2$-gradient flow of the functional $\SW_{\lambda_1,\lambda_2}$, which is the the sum of the Willmore energy, $\lambda_1$-weighted surface area, and $\lambda_2$-weighted enclosed volume, for surfaces immersed in $\R^3$. This coincides with the Helfrich functional with zero `spontaneous curvature'. Our first results are a concentration-compactness alternative and interior estimates for the flow. For initial data with small energy, we prove preservation of embeddedness, and by directly estimating the Euler-Lagrange operator from below in $L^2$ we obtain that the maximal time of existence is finite. Combining this result with the analysis of a suitable blowup allows us to show that for such initial data the flow contracts to a round point in finite time.

The Interior Euler-Lagrange Operator in Field Theory

Abstract The paper is devoted to the interior Euler-Lagrange operator in field theory, representing an important tool for constructing the variational sequence. We give a new invariant definition of this operator by means of a natural decomposition of spaces of differential forms, appearing in the sequence, which defines its basic properties. Our definition extends the well-known cases of the Euler-Lagrange class (Euler-Lagrange form) and the Helmholtz class (Helmholtz form). This linear operator has the property of a projector, and its kernel consists of contact forms. The result generalizes an analogous theorem valid for variational sequences over 1-dimensional manifolds and completes the known heuristic expressions by explicit characterizations and proofs.

A variational derivation of the geostrophic momentum approximation

AbstractThis paper demonstrates that the shallow water semigeostrophic equations arise from a degenerate second-order Hamilton principle of very special structure. The associated Euler–Lagrange operator factors into a fast and a slow first-order operator; restricting to the slow part yields the geostrophic momentum approximation as balanced dynamics. While semigeostrophic theory has been considered variationally before, this structure appears to be new. It leads to a straightforward derivation of the geostrophic momentum approximation and its associated potential vorticity law. Our observations further affirm, from a different point of view, the known difficulty in generalizing the semigeostrophic equations to the case of a spatially varying Coriolis parameter.

On application of Newton’s law to mechanical systems with motion constraints

This work is devoted to deriving and investigating conditions for the correct application of Newton’s law to mechanical systems subjected to motion constraints. It utilizes some fundamental concepts of differential geometry and treats both holonomic and nonholonomic constraints. This approach is convenient since it permits one to view the motion of any dynamical system as a path of a point on a manifold. In particular, the main focus is on the establishment of appropriate conditions, so that the form of Newton’s law of motion remains invariant when imposing an additional set of motion constraints on a mechanical system. Based on this requirement, two conditions are derived, specifying the metric and the form of the connection on the new manifold, which results after enforcing the additional constraints. The latter is weaker than a similar condition obtained by imposing a metric compatibility condition holding on Riemannian manifolds and employed frequently in the literature. This is shown to have several practical implications. First, it provides a valuable freedom for selecting the connection on the manifold describing large rigid body rotation, so that the group properties of this manifold are preserved. Moreover, it is used to state clearly the conditions for expressing Newton’s law on the tangent space and not on the dual space of a manifold, which is the natural geometrical space for this. Finally, the Euler–Lagrange operator is examined and issues related to equations of motion for anholonomic and vakonomic systems are investigated and clarified further.

Functional aspects of the Hardy inequality: appearance of a hidden energy

We revisit the classical Hardy inequality and the evolution problem associated to its Euler-Lagrange elliptic operator in the spirit of the paper by Vázquez and Zuazua (J Funct Analysis 173:103–153 2000). As a result, we establish the correct optimal functional setting by means of a reformulation of the problem that eliminates the possible indefinite character of the Hardy functional due to the singular behaviour of solutions near the origin. Surprisingly, the connection of the energy of the new formulation with the standard Hardy functional is nontrivial, due to the presence of a Hardy singularity energy. This corresponds to a loss for the total energy. The problem already arises when the equation is posed in a bounded domain, a case we study in full detail. We also consider an equivalent problem with inverse-square potential on an exterior domain. The extra energy term is then present as an effect that comes from infinity, a kind of hidden energy. In this case, in an unexpected way, this term is additive to the total energy, and it may even constitute the main part of it.

1-Soliton solution and conservation laws for nonlinear wave equation in semiconductors

This paper obtains the 1-soliton solution of a nonlinear wave equation that arises in the study of semiconductors. The conserved quantities are also calculated from this equation. Furthermore, additional non-trivial conserved quantities are computed using the invariance and multiplier approach based on the well known result that the Euler–Lagrange operator annihilates the total divergence.

1-Soliton solution and conservation laws for the Jaulent–Miodekequation with power law nonlinearity

This paper obtains the 1-soliton solution of the Jaulent–Miodek equation with power law nonlinearity. The solitary wave ansatz is used to obtain the soliton solution to this equation. Subsequently, the conserved quantities are computed using the invariance and multiplier approach based on the well known result that the Euler–Lagrange operator annihilates the total divergence.

Conserved current for the Cotton tensor, black hole entropy and equivariant Pontryagin forms

The Chern–Simons Lagrangian density in the space of metrics of a three-dimensional manifold M is not invariant under the action of diffeomorphisms on M. However, its Euler–Lagrange operator can be identified with the Cotton tensor, which is invariant under diffeomorphims. As the Lagrangian is not invariant, the Noether theorem cannot be applied to obtain conserved currents. We show that it is possible to obtain an equivariant conserved current for the Cotton tensor by using the first equivariant Pontryagin form on the bundle of metrics. Finally we define a Hamiltonian current which gives the contribution of the Chern–Simons term to the black hole entropy, energy and angular momentum.

The antifield Koszul–Tate complex of reducible Noether identities

A generic degenerate Lagrangian system of even and odd fields is examined in algebraic terms of the Grassmann-graded variational bicomplex. Its Euler–Lagrange operator obeys Noether identities which need not be independent, but satisfy first-stage Noether identities, and so on. We show that, if a certain necessary and sufficient condition holds, one can associate to a degenerate Lagrangian system the exact Koszul–Tate complex with the boundary operator whose nilpotency condition restarts all its Noether and higher-stage Noether identities. This complex provides a sufficient analysis of the degeneracy of a Lagrangian system for the purpose of its BV quantization.

NOETHER IDENTITIES OF A DIFFERENTIAL OPERATOR: THE KOSZUL–TATE COMPLEX

Given a generic Lagrangian system, its Euler–Lagrange operator obeys Noether identities which need not be independent, but satisfy first-stage Noether identities, and so on. This construction is generalized to arbitrary differential operators on a smooth fiber bundle. Namely, if a certain necessary and sufficient condition holds, one can associate to a differential operator the exact chain complex with the boundary operator whose nilpotency restarts all the Noether identities characterizing the degeneracy of an original differential operator.

Cartan forms for first order constrained variational problems

Given a constrained variational problem on the 1-jet extension J 1 Y of a fibre bundle p : Y → X , under certain conditions on the constraint submanifold S ⊂ J 1 Y , we characterize the space of admissible infinitesimal variations of an admissible section s : X → Y as the image by a certain first order differential operator, P s , of the space of sections Γ ( X , s ∗ V Y ) . In this way we obtain a constrained first variation formula for the Lagrangian density L ω on J 1 Y , which allows us to characterize critical sections of the problem as admissible sections s such that P s + E L ω ( s ) = 0 , where P s + is the adjoint operator of P s and E L ω ( s ) is the Euler–Lagrange operator of the Lagrangian density L ω as an unconstrained variational problem. We introduce a Cartan form on J 2 Y which we use to generalize the Cartan formalism and Noether theory of infinitesimal symmetries to the constrained variational problems under consideration. We study the relation of this theory with the Lagrange multiplier rule as well as the question of regularity in this framework. The theory is illustrated with several examples of geometrical and physical interest.

Variational formulation of Chern–Simons theory for arbitrary Lie groups

We show that the Chern–Simons theory for a principal G-bundle P over a three-dimensional manifold, with G an arbitrary Lie group, can be formulated as a variational problem defined by local data on the bundle of connections C(P) of P. By means of the theory of variational problems defined by local data we prove that the Euler–Lagrange operator and the differential of the Poincaré–Cartan form can be intrinsically expressed in terms of the symplectic form and the curvature morphism of C(P). These facts and the theory of the global inverse problem of the Calculus of Variations allow us to prove that there is indeed a global Lagrangian density for these theories. We also prove that every infinitesimal automorphism of P produces in a natural way an infinitesimal symmetry of the variational problem defined by the Chern–Simons theory. We therefore conclude that the algebra of infinitesimal symmetries of these theories is infinite dimensional.

First-order locally variational operators

First-order Lagrangian densities with first-order Euler–Lagrange operator are characterized. Variationally trivial Lagrangians and locally variational operators of first order are determined.