Lagrange Identity Research Articles

Обобщение тождества Лагранжа и новые интегралы движения

Обобщение тождества Лагранжа и новые интегралы движения

Lagrange’s identity and its generalizations

The famous Lagrange identity expresses the second derivative of the moment of inertia of a system of material points through the kinetic energy and homogeneous potential energy. The paper presents various extensions of this brilliant result to the case 1) of constrained mechanical systems, 2) when the potential energy is quasi-homogeneous in coordinates and 3) of continuum of interacting particles governed by the well-known Vlasov kinetic equation.

Тождество Лагранжа и его обобщения

The famous Lagrange identity expresses the second derivative of the moment of inertia of a system of material points through the kinetic energy and homogeneous potential energy. The paper presents various extensions of this brilliant result to the case 1) of constrained mechanical systems, 2) when the potential energy is quasi-homogeneous in coordinates and 3) of continuum of interacting particles governed by the well-known Vlasov kinetic equation.

Q-Dominant and q-recessive matrix solutions for linear quantum systems

In this study, linear second-order matrix q-difference equations are shown to be formally self-adjoint equations with respect to a certain inner prod- uct and the associated self-adjoint boundary conditions. A generalized Wronskian is introduced and a Lagrange identity and Abel's formula are established. Two reduction-of-order theorems are given. The analysis and characterization of q- dominant and q-recessive solutions at infinity are presented, emphasizing the case when the quantum system is disconjugate.

Conservative boundary control systems

We study continuous time linear dynamical systems of boundary control/observation type, satisfying a Green–Lagrange identity. Particular attention is paid to systems which have a well-defined dynamics both in the forward and the backward time directions. As we change the direction of time we also interchange inputs and outputs. We show that such a boundary control/observation system gives rise to a continuous time Livšic–Brodskiĭ (system) node with strictly unbounded control and observation operators. The converse is also true. We illustrate the theory by a classical example, namely, the wave equation describing the reflecting mirror.

Evaluating infinite range oscillatory integrals using generalised quadrature methods

The problem of evaluating infinite range irregularly oscillatory integrals is considered using the generalised quadrature approach based on Lagrange's identity. Some modifications are required beyond the method as applied to finite range problems and a series of practical examples are given. These split into two categories: those with exponentially decreasing weight functions and those without.

Universal Bounds on Coarsening Rates for Mean-Field Models of Phase Transitions

We prove one-sided universal bounds on coarsening rates for two kinds of mean-field models of phase transitions, one with a coarsening rate $l \sim t^{1/3}$ and the other with $l\sim t^{1/2}$. Here l is a characteristic length scale. These bounds are both proved by following a strategy developed by Kohn and Otto [ Comm. Math. Phys., 229 (2002), pp. 375--395]. The $l\sim t^{1/2}$ rate is proved using a new dissipation relation which extends the Kohn--Otto method. In both cases, the dissipation relations are subtle and their proofs are based on a residual lemma (Lagrange identity) for the Cauchy--Schwarz inequality.

The Structure of Gelfand-Levitan-Marhenko Type Equations for Delsarte Transmutation Operators of Linear Multidimensional Differential Operators and Operator Pencils. Part 1

An analog of Gelfand-Levitan-Marchenko integral equations for multi- dimensional Delsarte transmutation operators is constructed by means of studying their differential-geometric structure based on the classical Lagrange identity for a formally conjugated pair of differential operators. An extension of the method for the case of affine pencils of differential operators is suggested.

Improved bounds for some nonstandard problems in generalized heat conduction

We consider the Maxwell–Cattaneo system of equations for generalized heat conduction where the temperature and heat flux satisfy a nonstandard auxiliary condition which prescribes a combination of their values initially and at a later time. We obtain L 2 bounds for the temperature and heat flux by means of Lagrange identities. These bounds extend the range of validity for the parameter in the nonstandard condition under a constraint on the coefficients in the differential equations.

Classroom note: An extension of Lagrange's identity to matrices

In this note the well-known Lagrange identity is extended to matrices. The resulting generalized Lagrange identity is used to give characterizations of symmetry, commutativity of projectors and normality.

Structure of Binary Transformations of Darboux Type and Their Application to Soliton Theory

On the basis of generalized Lagrange identity for pairs of formally adjoint multidimensional differential operators and a special differential geometric structure associated with this identity, we propose a general scheme of the construction of corresponding transformation operators that are described by nontrivial topological characteristics. We construct explicitly the corresponding integro-differential symbols of transformation operators, which are used in the construction of Lax-integrable nonlinear two-dimensional evolutionary equations and their Darboux–Backlund-type transformations.

Some theoretical aspects of generalised quadrature methods

Generalised quadrature methods rely on generating quadrature rules for given irregular oscillatory weight functions w( x) commonly belonging to the class C n [ a, b], for some usually small n. If these weight functions are known to satisfy Lw=0 for a differential operator L, then Lagrange's identity gLw−wMg=Z′(w,g) can be used to generate a quadrature rule by forcing exactness for a set of basis functions. Theorems which give conditions under which the computed quadrature rules will yield results correct to a required precision (usually that of the machine being employed) underpin the practical rule, and finite range integrals with weights such as sin( q( x)) and J n ( q( x)) have been successfully integrated, for q( x)∈ C 2[ a, b]. Doubly oscillatory weights also become feasible with weights such as J n ( q 1( x)) J m ( q 2( x)). More recent work has considered multiple quadratures and the special problems which arise with the commonly occurring infinite range integrations. In the latter case, the direct approach results in violations of the conditions of the underlying theorem and requires some modification for success. This approach has enabled several diverse practical problems to be attempted including integrals from financial market predictions, from chemical reactor analysis, from coherent optical imaging and from wave analysis on sloping beaches.

Some remarks on growth and uniqueness in thermoelasticity

We use the Lagrange identity method and the logarithmic convexity to obtain uniqueness and exponential growth of solutions in the thermoelasticity of type III and thermoelasticity without energy dissipation. As this is not the first contribution of this kind in this theory, it is worth remarking that the assumptions we use here are different from those used in other previous contributions. We assume that the elasticity tensor is positive semidefinite, but we allow that the constitutive tensor of the entropy flux vector (kij), which is a characteristic tensor in this theory, is not sign‐definite. The Lagrange identity method is used to obtain uniqueness in the context of the thermoelasticity of type III. The fundamental key to obtain exponential growth in the thermoelasticity without energy dissipation is the use of a new functional. This functional is inspired in that it is used when the elasticity tensor is not sign‐definite, but (kij) is positive definite.

A Unified Approach to Linear Differential Algebraic Equations and their Adjoints

Instead of a single matrix occurring in the standard setting, the leading term of the linear differential algebraic equation is composed of a pair of well matched matrices. An index notion is proposed for the equations. The coefficients are assumed to be continuous and only certain subspaces have to be continuously differentiable. The solvability of lower index problems is proved. The solution representations are based on the solutions of certain inherent regular ordinary differential equations that are uniquely determined by the problem data. The assumptions allow for a unified treatment of the original equation and its adjoint. Both equations have the same index and are solvable simultaneously. Their fundamental solution matrices satisfy a relation that generalizes the classical Lagrange identity.

Some uniqueness and continuous dependence results in the theory of swelling porous elastic soils

This paper is concerned with the isothermal linear theory of swelling porous elastic soils. Initial-boundary value problems are formulated for the linear dynamic theory of an isothermal mixture consisting of three components: an elastic solid, a viscous fluid and a gas. Then the uniqueness and continuous dependence problems are discussed in connection with the solutions of such initial-boundary value problems. The uniqueness results are established under mild positive semi-definiteness assumptions or with no definiteness assumptions upon the internal energy. Various estimates are established for describing the continuous dependence of solutions with respect to the external given data. In this aim the Lagrange identity and the logarithmic convexity methods are used.

Uniqueness in exterior domains for the generalized heat conduction

In this note, we prove uniqueness of those solutions of the generalized heat conduction equation that increase as an exponential of the square of the distance from the origin. Continuous dependence results with respect to initial data and supply terms are also proved. The results are obtained with the help of the weighted energy method. We also prove uniqueness of the solutions of the backward in time problem. This second result is obtained by means of the weighted Lagrange identities method.

On the linear problem of swelling porous elastic soils

In this paper we consider the linear theory of swelling porous elastic soils. The formulation belongs to the theory of mixtures for porous elastic solids filled with fluid and gas. It proposes some new mathematical difficulties. Logarithmic convexity and/or the Lagrange identity method is used in the case of fluid and gas saturation. Continuous results dependent on initial conditions and supply terms are obtained in the general case. Spatial decay estimates are also obtained by means of comparison arguments. This last result is only valid when suitable conditions on the viscosity coefficients are satisfied.

Exactly integrable models of the wave interaction with continuous spectrum

Interaction with a continuous spectrum in inhomogeneous media is constructed the Laks representation of exactly integrable model of the wave by the generalized Lagrange identities method for the conjugated equations. The general analysis of the obtained equations and ways of their use in theoretical and applied physical problems of non-linear wave processes are carried out.

Uniqueness and continuous data dependence in the elastic cylinder

Selected continuous data dependence is established for the linearised elastic cylinder under relaxed positive-definite and homogeneity conditions on elasticities in the superposed linear deformation and for constrained sets of the solution. The proofs are based on those in [Math. Proc. Cambridge Philos. Soc. 103 (1988) 535] and employ a Lagrange identity obtained from Betti's reciprocal theorem. The treatment is intended only to be indicative of the method and is restricted to dependence on body force, certain base data, and the superposed elasticities in the half-length of the finite cylinder. Estimates are also presented for the semi-infinite cylinder under alternative boundary conditions. Uniqueness of the solution is also established.

83.67 A simple proof of the Lagrange identity on vector products

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