Variational sequences in mechanics

Let be a fibered manifold over a base manifold . A differential 1-form , defined on the -jet prolongation of , is said to be contact, if it vanishes along the -jet prolongation of every section of . The notion of contactness is naturally extended to -forms with . The contact forms define a subsequence of the De Rham sequence on . The corresponding quotient sequence is known as the rth order variational sequence. In this paper, the case of 1-dimensional base is considered. A simple proof is given of the fact that the rth order variational sequence is an acyclic resolution of the constant sheaf. Then the 1st order variational sequence is studied in detail. The quotient sheaves, as well as the quotient mappings, are determined explicitly, and their relationship to the standard concepts of the 1st order calculus of variations is discussed. The following is shown: a) the lagrangians in the 1st order variational sequence (classes of 1-forms) coincide with 2nd order lagrangians, affine in the second derivative variables, b) the concept of the Euler-Lagrange form is extended to 2-forms which are not necessarily variational, c) the concept of the Helmholtz-Sonin form is introduced as the class of an arbitrary 3-form, d) the well-known fundamental notions such as the Euler-Lagrange, and Helmholtz-Sonin mappings are represented by two arrows at the beginning of the variational sequence; this relates the global structure of the Euler-Lagrange mapping to the cohomology of , e) all the remaining classes of -forms with , as well as the quotient mappings, are determined explicitly, f) a locally variational form is defined as a generalization of a symplectic form; locally variational forms, associated to a fixed Euler-Lagrange form, are characterized, and g) distributions associated with a locally variational form are described, and their relation to the Euler-Lagrange equations is studied. These results illustrate differences between finite order variational sequences and variational bicomplexes, based on infinite jet constructions. Received February 18, 1996 / In revised form December 1996 / Accepted December 2, 1996     

Computing ODE symmetries as abnormal variational symmetries

We give a new computational method to obtain symmetries of ordinary differential equations. The proposed approach appears as an extension of a recent algorithm to compute variational symmetries of optimal control problems [P.D.F. Gouveia, D.F.M. Torres, Automatic computation of conservation laws in the calculus of variations and optimal control, Comput. Methods Appl. Math. 5 (4) (2005) 387-409], and is based on the resolution of a first order linear PDE that arises as a necessary and sufficient condition of invariance for abnormal optimal control problems. A computer algebra procedure is developed, which permits one to obtain ODE symmetries by the proposed method. Examples are given, and results compared with those obtained by previous available methods.     

Nonholonomic problems and the theory of distributions

This article constitutes an appendix to the book by P. A. Griffiths, Exterior differential systems and the calculus of variations. Birkhäuser, 1983. It especially focusses on the distinction between holonomic and nonholonomic mechanical and variational problems, and indicates how rich and interesting the phenomena are in the nonholonomic case.Appendix to the Russian translation of [G]. The letter G followed by a number will refer to the corresponding reference in [G]; the quoted references, as well as [G] itself, can be found at the end of the list of references below.     

Strict feasibility and stable solvability of bifunction variational inequalities

In this paper, we study strict feasibility of a bifunction variational inequality. It is proved that a monotone bifunction variational inequality has a nonempty and bounded solution set if and only if it is strictly feasible. Stable solvability of the bifunction variational inequality is discussed under strict feasibility assumption when the domain set is perturbed. Our results generalize earlier results on the classical variational inequality to the case of the bifunction variational inequality.     

Reduction and Equivariant Branching Lemma: Dynamical systems,evolution PDEs,and gauge theories

We review and recast the Equivariant Branching Lemma-which has proved a remarkable tool in linearly equivariant bifurcation theory-and consider its extension to the case of nonlinear (Lie-point) symmetries. This is then applied to gauge theories and gauge theoretic problems, and to nonlinear evolution PDE's; the paper also contains an original setting of Lie-point symmetries for evolution PDEs, modelled on the dynamical systems setting.     

Contact Symmetries and Variational PDE’s

Symmetries of the contact ideal on the r-jet bundle over a fibred manifold are studied, and transformation properties under contact symmetries of different objects in the variational sequence related with systems of partial differential equations are investigated. This paper is dedicated to Valentin Lychagin on the occasion of his 60th birthday.     

Hypersurfaces of prescribed mean curvature in central projection-II

This work, which is a direct continuation of a preceding paper, is dedicated to interior regularity results with respect to a singular variational integral with linear growth. The Dirichlet Problem associated with the corresponding variational problem describes surfaces of prescribed mean curvature that admits of a central projection onto an open domain of the unit sphere Sⁿ. Our results are accomplished by using methods from geometric measure theory together with certain L bounds that we shall prove here as well.Received: 11 February 2003     

Nonconvex differential calculus for infinite-dimensional multifunctions

The paper is concerned with generalized differentiation of set-valued mappings between Banach spaces. Our basic object is the so-called coderivative of multifunctions that was introduced earlier by the first author and has had a number of useful applications to nonlinear analysis, optimization, and control. This coderivative is a nonconvex-valued mapping which is related to sequential limits of Fréchet-like graphical normals but is not dual to any tangentially generated derivative of multifunctions. Using a variational approach, we develop a full calculus for the coderivative in the framework of Asplund spaces. The latter class is sufficiently broad and convenient for many important applications. Some useful calculus results are also obtained in general Banach spaces.This research was partially supported by the National Science Foundation under grants DMS-9206989 and DMS-9404128, by the USA-Israel grant 94-00237, and by the NATO contract CRG-950360.     

LEGENDRE TRANSFORMATIONS AND CARTAN FORMS IN THE CALCULUS OF VARIATIONS OF MULTIPLE INTEGRALS

ABSTRACT

(PART I): A field-theoretic treatment of variational problems in n independent variables {x^j} and N dependent variables {ψ^A)} is presented that differs substantially from the standard field theories, such as those of Carathéodory [4] and Weyl [10], inasmuch as it is not stipulated ab initio that the Lagrangian be everywhere positive. This is accomplished by the systematic use of a canonical formalism. Since the latter must necessarily be prescribed by appropriate Legendre transformations, the construction of such transformations is the central theme of Part I.—The underlying manifold is M = M_n x M_N, where M_n, M_N are manifolds with local coordinates {x^j}, {ψ^A}, respectively. The basic ingredient of the theory consists of a pair of complementary distributions D_n, D_N on M that are defined respectively by the characteristic subspaces in the tangent spaces of M of two sets of smooth 1-forms {π^A:A = 1,…, N}, {π^j = 1,…, n} on M. For a given local coordinate system on M the planes of 4, have unique (adapted) basis elements B_j = (?/?_x ^j) + B^A _j (?/?ψ^A), whose coefficients B^A _j will assume the role of derivatives such as ?ψ^A/?x^j in the final analysis of Part II. The first step toward a Legendre transformation is a stipulation that prescribes B^A _j as a function of the components {π^j _h,π^j _A} of {π^j}—these components being ultimately the canonical Variables—in such a manner that B^A _j is unaffected by the action of any unimodular transformation applied to the exterior system {π^j}. A function H of the canonical variables is said to be an acceptable Hamiltonian if it satisfies a similar invariance requirement, together with a determinantal condition that involves its Hessian with respect to π^j _A. The second part of the Legendre transformation consists of the identification in terms of H and the canonical variables of a function L that depends solely on B^A _j and the coordinates on M. This identification imposes a condition on the Hessian of L with respect to B^A _j. Conversely, any function L that satisfies these requirements is an acceptable Lagrangian, whose Hamiltonian is uniquely determined by the general construction.     

A method for computing symmetries and conservation laws of integra-differential equations

A method for computing symmetries and conservation laws of integro-differential equations is proposed. It resides in reducing an integro-differential equation to a system of boundary differential equations and in computing symmetries and conservation laws of this system. A geometry of boundary differential equations is constructed like the differential case. Results of the computation for the Smoluchowski's coagulation equation are given.     

Strict approximate solutions in set-valued optimization with applications to the approximate Ekeland variational principle

This work deals with strict solutions of set-valued optimization problems under the set optimality criterion. In this context, we introduce a new approximate solution concept and we obtain several properties of these solutions when the error is fixed and also for their limit behavior when the error tends to zero. Then we prove a general existence result, which is applied to obtain approximate Ekeland variational principles.     

Nonlocal symmetries and the theory of coverings: An addendum to A. M. vinogradov's ‘local symmetries and conservation laws”

For a systemY of partial differential equations, the notion of a covering Y is introduced whereY is infinite prolongation ofY. Then nonlocal symmetries ofY are defined as transformations of which conserve the underlying contact structure. It turns out that generating functions of nonlocal symmetries are integro-differential-type operators.     

Solvability of dynamic antiplane frictional contact problems for viscoelastic cylinders

We study a mathematical model which describes the antiplane shear deformations of a cylinder in frictional contact with a rigid foundation. The process is dynamic, the material behavior is described with a linearly viscoelastic constitutive law and friction is modeled with a general subdifferential boundary condition. We derive a variational formulation of the model which is in a form of an evolutionary hemivariational inequality for the displacement field. Then we prove the existence of a weak solution to the model. The proof is based on an abstract result for second order evolutionary inclusions in Banach spaces. Also, we prove that, under additional assumptions, the weak solution to the model is unique. We complete our results with concrete examples of friction laws for which our results are valid.     

Symmetries of compatibility conditions for systems of differential equations

When symmetries of differential equations are applied, various types of associated systems of equations appear. Compatibility conditions of the associated systems expressed in the form of differential equations inherit Lie symmetries of the initial equations. Invariant solutions to compatibility systems are known as orbits of partially invariant and generic solutions involved in the Lie group foliation of differential equations and so on. In some cases Bäcklund transformations and differential substitutions connecting quotient equations for compatibility conditions and initial systems naturally arise. Besides, Ovsiannikov's orbit method for finding partially invariant solutions is essentially based on such symmetries.     

Analysis of a dynamic contact problem for electro-viscoelastic cylinders

We consider a mathematical model which describes the antiplane shear deformations of a piezoelectric cylinder in frictional contact with a foundation. The process is mechanically dynamic and electrically static, the material behavior is described with a linearly electro-viscoelastic constitutive law, the contact is frictional and the foundation is assumed to be electrically conductive. Both the friction and the electrical conductivity condition on the contact surface are described with subdifferential boundary conditions. We derive a variational formulation of the problem which is of the form of a system coupling a second order hemivariational inequality for the displacement field with a time-dependent hemivariational inequality for the electric potential field. Then we prove the existence of a unique weak solution to the model. The proof is based on abstract results for second order evolutionary inclusions in Banach spaces. Finally, we present concrete examples of friction laws and electrical conductivity conditions for which our result is valid.     

Symmetries and conservation laws of partial differential equations: Basic notions and results

The main notions and results which are necessary for finding higher symmetries and conservation laws for general systems of partial differential equations are given. These constitute the starting point for the subsequent papers of this volume. Some problems are also discussed.     

Nonlocal trends in the geometry of differential equations: Symmetries,conservation laws,and Bäcklund transformations

The theory of coverings over differential equations is exposed which is an adequate language for describing various nonlocal phenomena: nonlocal symmetries and conservation laws, Bäcklund transformations, prolongation structures, etc. A notion of a nonlocal cobweb is introduced which seems quite useful for dealing with nonlocal objects.     

On the thread problem for minimal surfaces

For a given one-dimensional fixed boundary $\Gamma$ in and a given constant we consider any one-dimensional free boundary $F$ in subject to the conditions that the length of is equal to , that and form a closed boundary, and that the minimal surface of dimension two being bounded by and minimizes the area among all comparison surfaces being bounded by and some with length equal to . This variational problem is known as the thread problem for minimal surfaces and stems from soap film experiments, in which the fixed boundary parts are pieces of wires and the free boundary parts are threads. The new result of this article will be that has no singular points in , provided the admissible surfaces and boundary parts are supposed to be rectifiable flat chains modulo two. Received February 16, 1995 / Accepted October 20, 1995